1 | ;;; -*- Mode: Lisp; Syntax: Common-Lisp; Package: Grobner; Base: 10 -*-
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2 | #|
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3 | $Id: grobner.lisp,v 1.8 2009/01/23 10:40:26 marek Exp $
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4 | *--------------------------------------------------------------------------*
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5 | | Copyright (C) 1994, Marek Rychlik (e-mail: rychlik@math.arizona.edu) |
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6 | | Department of Mathematics, University of Arizona, Tucson, AZ 85721 |
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7 | | |
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8 | | Everyone is permitted to copy, distribute and modify the code in this |
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9 | | directory, as long as this copyright note is preserved verbatim. |
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10 | *--------------------------------------------------------------------------*
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11 | |#
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12 | (defpackage "GROBNER"
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13 | (:use "ORDER" "PARSE" "PRINTER" "DIVISION" "MONOM"
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14 | "MAKELIST" "COEFFICIENT-RING" "TERM" "POLY"
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15 | "POLY-WITH-SUGAR" "COMMON-LISP")
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16 | (:export
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17 | spoly
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18 | grobner
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19 | reduction
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20 | normal-form
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21 | reduced-grobner
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22 | normalize-poly
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23 | normalize-basis
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24 | elimination-ideal
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25 | ring-intersection
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26 | poly-contract
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27 | ideal-intersection
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28 | poly-lcm
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29 | grobner-gcd
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30 | pseudo-divide
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31 | buchberger-criterion
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32 | grobner-test
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33 | grobner-equal
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34 | grobner-subsetp
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35 | grobner-member
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36 | ideal-equal
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37 | ideal-subsetp
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38 | ideal-member
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39 | ideal-saturation
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40 | ideal-polysaturation
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41 | ideal-saturation-1
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42 | ideal-polysaturation-1
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43 | grobner-primitive-part
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44 | grobner-content
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45 | colon-ideal
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46 | *grobner-debug*
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47 | buchberger-set-pair-heuristic
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48 | gebauer-moeller-set-pair-heuristic
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49 | select-grobner-algorithm
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50 | spoly-with-sugar
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51 | normal-form-with-sugar
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52 | grobner-with-sugar
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53 | ))
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54 |
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55 | (in-package "GROBNER")
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56 |
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57 | #+debug(proclaim '(optimize (speed 0) (debug 3)))
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58 | #-debug(proclaim '(optimize (speed 3) (debug 0)))
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59 |
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60 | #+debug
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61 | (defvar *grobner-debug* nil
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62 | "If T, produce debugging and tracing output.")
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63 |
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64 | (defvar *buchberger-merge-pairs* 'buchberger-merge-pairs-smallest-lcm
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65 | "A variable holding the predicate used to sort critical pairs
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66 | in the order of decreasing priority for the Buchberger algorithm.")
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67 |
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68 | (defvar *gebauer-moeller-merge-pairs* 'gebauer-moeller-merge-pairs-use-mock-spoly
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69 | "A variable holding the predicate used to sort critical pairs
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70 | in the order of decreasing priority for the Gebauer-Moeller algorithm")
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71 |
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72 | (defvar *grobner-function* 'buchberger
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73 | "The name of the function used to calculate Grobner basis")
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74 |
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75 | (defun select-grobner-algorithm (algorithm)
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76 | "Select one of the several implementations of the Grobner basis
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77 | algorithm."
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78 | (setf *grobner-function*
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79 | (ecase algorithm
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80 | (:buchberger 'buchberger)
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81 | (:cached-buchberger 'cached-buchberger)
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82 | (:gebauer-moeller 'gebauer-moeller)
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83 | (:sugar 'buchberger-with-sugar)
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84 | (:cached-sugar 'cached-buchberger-with-sugar)
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85 | (:gebauer-moeller-with-sugar 'gebauer-moeller-with-sugar))))
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86 |
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87 | (defun grobner (F &optional (pred #'lex>)
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88 | (start 0)
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89 | (top-reduction-only nil)
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90 | (ring *coefficient-ring*))
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91 | "Return Grobner basis of an ideal generated by polynomial list F.
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92 | Assume that the monomials of each polynomial are ordered according to
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93 | the admissible monomial order PRED. The result will be a list of
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94 | polynomials ordered as well according to PRED. The polynomials from 0
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95 | to START-1 in the list F are assumed to be a Grobner basis, and thus
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96 | certain critical pairs do not have to be calculated. If
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97 | TOP-REDUCTION-ONLY is not NIL then only top reductions will be
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98 | performed, instead of full division, and thus the returned Grobner
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99 | basis will have its polynomials only top-reduced with respect to the
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100 | preceding polynomials. RING should be set to the RING structure (see
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101 | COEFFICIENT-RING package) corresponding to the coefficients of the
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102 | polynomials in the list F. This function is currently just an
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103 | interface to several algorithms which fine Grobner bases. Use
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104 | SELECT-GROBNER-ALGORITHM in order to change the algorithm."
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105 | (funcall *grobner-function* F pred start top-reduction-only ring))
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106 |
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107 |
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108 | #+debug
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109 | (defmacro debug-cgb (&rest args)
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110 | "This macro is used in printing debugging and tracing messages
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111 | and its arguments are analogous to the FORMAT function, except
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112 | that the stream argument is omitted. By default, the output is
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113 | sent to *trace-output*, which can be set to produce output
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114 | in a file."
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115 | `(when *grobner-debug* (format *trace-output* ,@args)))
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116 |
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117 | (defun spoly (f g pred ring)
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118 | "Return the S-polynomial of F and G, which should
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119 | be two polynomials with terms ordered by PRED and
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120 | coefficients in a ring whose operations are the slots
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121 | of the RING structure."
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122 | (declare (optimize (speed 3) (safety 0)))
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123 | (let* ((lcm (monom-lcm (lm f) (lm g)))
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124 | (m1 (monom/ lcm (lm f)))
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125 | (m2 (monom/ lcm (lm g))))
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126 | (let* ((c (funcall (ring-lcm ring) (lc f) (lc g)))
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127 | (cf (funcall (ring-/ ring) c (lc f)))
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128 | (cg (funcall (ring-/ ring) c (lc g))))
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129 | (poly-
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130 | (term-times-poly (cons m1 cf) (rest f) ring)
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131 | (term-times-poly (cons m2 cg) (rest g) ring)
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132 | pred
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133 | ring))))
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134 |
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135 |
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136 | (defun grobner-primitive-part (p ring)
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137 | "Divide polynomial P with integer coefficients by gcd of its
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138 | coefficients and return the result. Use the RING structure from the
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139 | COEFFICIENT-RING package to perform the operations on the
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140 | coefficients."
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141 | (if (poly-zerop p)
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142 | (values p 1)
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143 | (let ((c (grobner-content p ring)))
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144 | (values (mapcar
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145 | #'(lambda (x)
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146 | (cons (car x)
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147 | (funcall
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148 | (ring-/ ring)
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149 | (cdr x) c)))
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150 | p)
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151 | c))))
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152 |
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153 |
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154 | (defun grobner-content (p ring)
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155 | "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
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156 | to compute the greatest common divisor."
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157 | (let ((c (apply (ring-gcd ring) (mapcar #'cdr p))))
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158 | (when (/= (funcall (ring-signum ring) (cdar p))
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159 | (funcall (ring-signum ring) c))
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160 | (setf c (funcall (ring-- ring) c)))
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161 | c))
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162 |
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163 | (defun normal-form (f fl pred top-reduction-only ring)
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164 | "Calculate the normal form of the polynomial F with respect to
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165 | the polynomial list FL. Destructive to F; thus, copy-tree should be used
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166 | where F needs to be preserved. It returns multiple values.
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167 | The first value is the polynomial R which is reduced (or top-reduced
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168 | if TOP-REDUCTION-ONLY is not NIL). The second value is an integer
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169 | which is the number of reductions actually performed. The third value
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170 | is the coefficient by which F needs to be multiplied in order to be
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171 | able to perform the division without actually having to divide in the
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172 | coefficient ring (a form of pseudo-division is used). All operations
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173 | on the coefficients are performed using the RING structure from the
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174 | COEFFICIENT-RING package."
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175 | (declare (optimize (speed 3) (safety 0)))
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176 | ;; Loop invariant: c*f=sum ai*fi+r+p
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177 | (do (r (c (ring-unit ring))
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178 | (division-count 0)
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179 | (p f))
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180 | ((or (endp p)
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181 | (endp fl)
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182 | (and top-reduction-only r))
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183 | #+debug(progn
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184 | (debug-cgb "~&~3T~d reduction~:p" division-count)
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185 | (when (endp r)
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186 | (debug-cgb " ---> 0")))
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187 | (values (append (nreverse r) p)
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188 | division-count c))
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189 | (declare (fixnum division-count) (integer c))
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190 | (let ((g (find (lm p) fl :test #'monom-divisible-by-p :key #'lm)))
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191 | (cond
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192 | (g ;division possible
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193 | (incf division-count)
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194 | (let* ((gcd (funcall (ring-gcd ring) (lc g) (lc p)))
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195 | (c1 (funcall (ring-/ ring) (lc g) gcd))
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196 | (c2 (funcall (ring-/ ring) (lc p) gcd))
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197 | (m (monom/ (lm p) (lm g))))
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198 | ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
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199 | (setf r (scalar-times-poly c1 r ring)
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200 | c (funcall (ring-* ring) c c1)
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201 | p (grobner-op c2 c1 m (rest p) (rest g) pred ring))))
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202 | (t ;no division possible
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203 | (setf r (cons (lt p) r) ;move lt(p) to remainder
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204 | p (rest p))))))) ;remove lt(p) from p
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205 |
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206 | (defun buchberger (F pred start top-reduction-only ring
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207 | &aux (s (1- (length F))) B M)
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208 | "An implementation of the Buchberger algorithm. Return Grobner
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209 | basis of the ideal generated by the polynomial list F.
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210 | The terms of each polynomial are ordered by the admissible
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211 | monomial order PRED. Polynomials 0 to START-1 are assumed to
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212 | be a Grobner basis already, so that certain critical pairs
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213 | will not be examined. If TOP-REDUCTION-ONLY set, top reduction
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214 | will be preformed. The RING structure is used to perform operations
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215 | on coefficents (see COEFFICIENT-RING package)."
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216 | (declare (fixnum s))
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217 | (unless (plusp s) (return-from buchberger F)) ;cut startup costs
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218 | #+debug (debug-cgb "~&GROBNER BASIS - BUCHBERGER ALGORITHM")
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219 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
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220 | #+grobner-check (when (plusp start)
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221 | (grobner-test (subseq F 0 start) (subseq F 0 start) pred ring))
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222 | (setf B (nconc (makelist (list i j) (i 0 (1- start)) (j start s))
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223 | (makelist (list i j) (i start (1- s)) (j (1+ i) s)))
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224 | M (make-hash-table :test #'equal)
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225 | B (buchberger-sort-pairs B F pred ring))
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226 | ;;Initialize treated pairs
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227 | (dotimes (i (1- start))
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228 | (do ((j (1+ i) (1+ j))) ((>= j start))
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229 | (setf (gethash (list i j) M) t)))
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230 | (do ((G F))
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231 | ((endp B)
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232 | #+grobner-check(grobner-test G F pred ring)
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233 | #+debug(debug-cgb "~&GROBNER END")
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234 | G)
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235 | (let ((pair (pop B)))
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236 | (unless (or (Criterion-1 pair G)
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237 | (Criterion-2 pair G M))
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238 | (let ((SP (normal-form (spoly (elt G (first pair))
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239 | (elt G (second pair))
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240 | pred
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241 | ring)
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242 | G pred top-reduction-only ring)))
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243 | (unless (poly-zerop SP) ;S-polynomial ---> 0;
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244 | (setf s (1+ s)
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245 | SP (grobner-primitive-part SP ring)
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246 | G (nconc G (list SP))
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247 | B (funcall *buchberger-merge-pairs* B (makelist (list i s)
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248 | (i 0 (1- s)))
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249 | G pred ring))
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250 | #+debug (debug-cgb "~&Polynomials: ~d; Pairs left: ~d; Treated pairs: ~d"
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251 | (length G) (length B)
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252 | (hash-table-count M)))))
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253 | (setf (gethash pair M) t))))
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254 |
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255 | (defun grobner-op (c1 c2 m A B pred ring &aux n A1 A2)
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256 | "Perform a calculation equivalent to
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257 | (POLY- (SCALAR-TIMES-POLY C2 A)
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258 | (SCALAR-TIMES-POLY C1 (MONOM-TIMES-POLY M B))
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259 | PRED)
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260 | but more efficiently; it destructively modifies A and stores the
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261 | result in it; A is returned."
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262 | (dolist (x A) (setf (cdr x) (funcall (ring-* ring) c2 (cdr x))))
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263 | (unless B
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264 | (return-from grobner-op A))
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265 | (unless A
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266 | (return-from grobner-op
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267 | (mapcar
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268 | #'(lambda (x) (cons (monom* m (car x))
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269 | (funcall (ring-- ring)
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270 | (funcall (ring-* ring)
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271 | c1 (cdr x)))))
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272 | B)))
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273 | (setf A1 (cons nil A) A2 A1)
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274 | (tagbody
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275 | begin-outer-loop
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276 | (unless B
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277 | (return-from grobner-op (cdr A2)))
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278 | (setf n (monom* m (caar B)))
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279 | (tagbody
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280 | begin-inner-loop
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281 | (unless (cdr A1)
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282 | (setf (cdr A1)
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283 | (mapcar #'(lambda (x) (cons (monom* m (car x))
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284 | (funcall (ring-- ring)
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285 | (funcall (ring-* ring)
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286 | c1 (cdr x))))) B))
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287 | (return-from grobner-op (cdr A2)))
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288 | (multiple-value-bind (lm-greater lm-equal)
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289 | (funcall pred (caadr A1) n)
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290 | (cond
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291 | (lm-equal
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292 | (let ((s (funcall (ring-- ring)
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293 | (cdadr A1)
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294 | (funcall (ring-* ring) c1 (cdar B)))))
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295 | (if (funcall (ring-zerop ring) s) ;check for cancellation
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296 | (setf (cdr A1) (cddr A1))
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297 | (setf (cdadr A1) s))
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298 | (setf B (cdr B))
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299 | (go begin-outer-loop)))
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300 | (lm-greater
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301 | (setf A1 (cdr A1))
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302 | (go begin-inner-loop))
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303 | (t ;(lm B) is greater than (lm A)
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304 | (setf (cdr A1)
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305 | (cons
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306 | (cons
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307 | n
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308 | (funcall (ring-- ring)
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309 | (funcall (ring-* ring) c1 (cdar B))))
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310 | (cdr A1))
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311 | B (cdr B)
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312 | A1 (cdr A1))
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313 | (go begin-outer-loop)))))))
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314 |
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315 | ;;----------------------------------------------------------------
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316 | ;; Pairs
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317 | ;;----------------------------------------------------------------
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318 |
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319 | (defun buchberger-sort-pairs (B G pred ring)
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320 | "Sort critical pairs B by the function which is
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321 | the value of the vairable *buchberger-merge-pairs*.
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322 | G is the list of polynomials which the elemnts
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323 | of B point into. PRED is the admissible monomial order.
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324 | The RING structure holds operations on the coefficients
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325 | as described in the COEFFICIENT-RING package."
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326 | (funcall *buchberger-merge-pairs* nil B G pred ring))
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327 |
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328 |
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329 | (defun mock-spoly (f g pred)
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330 | "Returns an upper-bound on the leading
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331 | monomial of the S-polynomial of F and G, which are two polynomials
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332 | sorted by the admissible monomial order PRED."
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333 | (let* ((lcm (monom-lcm (lm f) (lm g)))
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334 | (m1 (monom/ lcm (lm f)))
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335 | (m2 (monom/ lcm (lm g))))
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336 | (cond
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337 | ((endp (second f))
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338 | (monom* m2 (car (second g))))
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339 | ((endp (second g))
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340 | (monom* m1 (car (second f))))
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341 | (t (let ((n1 (monom* m2 (car (second g))))
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342 | (n2 (monom* m1 (car (second f)))))
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343 | (if (funcall pred n1 n2)
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344 | n1
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345 | n2))))))
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346 |
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347 | ;;----------------------------------------------------------------
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348 | ;; A stragegy based on a predicted approximate S-polynomials
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349 | ;;----------------------------------------------------------------
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350 | (defun buchberger-merge-pairs-use-mock-spoly (B C G pred ring)
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351 | "Merges lists of critical pairs B and C into one list.
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352 | The pairs are assumed to be ordered according to the
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353 | increasing value of MOCK-SPOLY, as determined by the admissible
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354 | monomial order PRED. G is the list of polynomials which the pairs
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355 | of integers in B and C point into, and RING is the ring structure
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356 | defined in the COEFFICIENT-RING package."
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357 | (declare (ignore ring))
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358 | ;; Delete pairs consisting of two single monomials
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359 | (setf C (delete-if
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360 | #'(lambda (p) (and (endp (cdr (elt G (first p))))
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361 | (endp (cdr (elt G (second p))))))
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362 | C))
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363 | (labels ((pair-key (p) (mock-spoly (elt G (first p))
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364 | (elt G (second p)) pred))
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365 | (pair-order (x y) (funcall pred y x)))
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366 | (merge 'list B (sort C #'pair-order :key #'pair-key)
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367 | #'pair-order :key #'pair-key)))
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368 |
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369 |
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370 | ;;----------------------------------------------------------------
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371 | ;; A strategy based on the smallest lcm of the leading
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372 | ;; monomials of the two polynomials - so called normal strategy
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373 | ;;----------------------------------------------------------------
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374 | (defun buchberger-merge-pairs-smallest-lcm (B C G pred ring)
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375 | "Merges lists of critical pairs B and C into a single ordered
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376 | list. Implements a strategy of ordering pairs according to the
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377 | smallest lcm of the leading monomials of the two polynomials - so
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378 | called normal strategy."
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379 | (declare (ignore ring))
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380 | (labels ((pair-key (p) (monom-lcm (lm (elt G (first p)))
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381 | (lm (elt G (second p)))))
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382 | (pair-order (x y) (funcall pred y x)))
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383 | (merge 'list B (sort C #'pair-order :key #'pair-key)
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384 | #'pair-order :key #'pair-key)))
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385 |
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386 | ;;----------------------------------------------------------------
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387 | ;; A strategy based on the smallest degree of the lcm of the leading
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388 | ;; monomials of the two polynomials
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389 | ;;----------------------------------------------------------------
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390 | (defun buchberger-merge-pairs-use-smallest-degree (B C G pred ring)
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391 | "Mergest lists B and C of critical pairs. This strategy is based on
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392 | ordering the pairs according to the smallest degree of the lcm of the
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393 | leading monomials of the two polynomials."
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394 | (declare (ignore pred ring))
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395 | (labels ((pair-key (p) (total-degree
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396 | (monom-lcm (lm (elt G (first p)))
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397 | (lm (elt G (second p)))))))
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398 | (merge 'list B (sort C #'< :key #'pair-key) #'< :key #'pair-key)))
|
---|
399 |
|
---|
400 | (defun buchberger-merge-pairs-use-smallest-length (B C G pred ring)
|
---|
401 | "Mergest lists B and C of critical pairs. This strategy is based on
|
---|
402 | ordering the pairs according to the smallest total length of the
|
---|
403 | two polynomials, where length is the number of terms."
|
---|
404 | (declare (ignore pred ring))
|
---|
405 | (labels ((pair-key (p) (+
|
---|
406 | (length (elt G (first p)))
|
---|
407 | (length (elt G (second p))))))
|
---|
408 | (merge 'list B (sort C #'< :key #'pair-key) #'< :key #'pair-key)))
|
---|
409 |
|
---|
410 | (defun buchberger-merge-pairs-use-smallest-coefficient-length (B C G pred ring)
|
---|
411 | "Mergest lists B and C of critical pairs. This strategy is based on
|
---|
412 | ordering the pairs according to the smallest combined length of the
|
---|
413 | coefficients of the two polynomials."
|
---|
414 | (declare (ignore pred))
|
---|
415 | (labels ((coefficient-length (poly)
|
---|
416 | (apply #'+ (mapcar (ring-length ring)
|
---|
417 | (mapcar #'cdr poly))))
|
---|
418 | (pair-key (p) (+
|
---|
419 | (coefficient-length (elt G (first p)))
|
---|
420 | (coefficient-length (elt G (second p))))))
|
---|
421 | (merge 'list B (sort C #'< :key #'pair-key) #'< :key #'pair-key)))
|
---|
422 |
|
---|
423 | (defun buchberger-set-pair-heuristic
|
---|
424 | (method
|
---|
425 | &aux
|
---|
426 | (strategy-fn
|
---|
427 | (ecase method
|
---|
428 | (:minimal-mock-spoly #'buchberger-merge-pairs-use-mock-spoly)
|
---|
429 | (:minimal-lcm #'buchberger-merge-pairs-smallest-lcm)
|
---|
430 | (:minimal-total-degree #'buchberger-merge-pairs-use-smallest-degree)
|
---|
431 | (:minimal-length #'buchberger-merge-pairs-use-smallest-length)
|
---|
432 | (:minimal-coefficient-length #'buchberger-merge-pairs-use-smallest-coefficient-length))))
|
---|
433 | "Simply sets the variable *buchberger-merge-pairs* to the heuristic
|
---|
434 | function STRATEGY-FN implementing one of several strategies introduces
|
---|
435 | before of selecting the most promising. METHOD should be one of the
|
---|
436 | listed keywords."
|
---|
437 | (setf *buchberger-merge-pairs* strategy-fn))
|
---|
438 |
|
---|
439 | ;;----------------------------------------------------------------
|
---|
440 | ;; Grobner Criteria
|
---|
441 | ;;----------------------------------------------------------------
|
---|
442 | (defun Criterion-1 (pair G &aux (i (first pair)) (j (second pair)))
|
---|
443 | "Returns T if the leading monomials of the two polynomials
|
---|
444 | in G pointed to by the integers in PAIR have disjoint (relatively prime)
|
---|
445 | monomials. This test is known as the first Buchberger criterion."
|
---|
446 | (monom-rel-prime (lm (elt G i))
|
---|
447 | (lm (elt G j))))
|
---|
448 |
|
---|
449 |
|
---|
450 | (defun Criterion-2 (pair G M &aux (i (first pair)) (j (second pair)))
|
---|
451 | "Returns T if the leading monomial of some element P of G divides
|
---|
452 | the LCM of the leading monomials of the two polynomials in the
|
---|
453 | polynomial list G, pointed to by the two integers in PAIR, and P
|
---|
454 | paired with each of the polynomials pointed to by the the PAIR has
|
---|
455 | already been treated, as indicated by the hash table M, which stores
|
---|
456 | value T for every treated pair so far."
|
---|
457 | (labels ((make-pair (u v) (if (< u v) (list u v) (list v u))))
|
---|
458 | (dotimes (k (length G) nil)
|
---|
459 | (when (and (/= k i)
|
---|
460 | (/= k j)
|
---|
461 | (monom-divides-p
|
---|
462 | (lm (elt G k))
|
---|
463 | (monom-lcm (lm (elt G i)) (lm (elt G j))))
|
---|
464 | (gethash (make-pair i k) M)
|
---|
465 | (gethash (make-pair k j) M))
|
---|
466 | #+debug(when *grobner-debug* (format t ":B2"))
|
---|
467 | (return-from Criterion-2 t)))))
|
---|
468 |
|
---|
469 | ;;----------------------------------------------------------------
|
---|
470 | ;; End of GROBNER implementation
|
---|
471 | ;;----------------------------------------------------------------
|
---|
472 |
|
---|
473 | (defun normalize-poly (p ring)
|
---|
474 | "Divide a polynomial by its leading coefficient. It assumes
|
---|
475 | that the division is possible, which may not always be the
|
---|
476 | case in rings which are not fields. The exact division operator
|
---|
477 | is assumed to be provided by the RING structure of the
|
---|
478 | COEFFICIENT-RING package."
|
---|
479 | (mapcar #'(lambda (term)
|
---|
480 | (cons
|
---|
481 | (car term)
|
---|
482 | (funcall (ring-/ ring) (cdr term) (lc p)))) p))
|
---|
483 |
|
---|
484 | (defun normalize-basis (plist ring)
|
---|
485 | "Divide every polynomial in a list PLIST by its leading coefficient.
|
---|
486 | Use RING structure to perform the division of the coefficients."
|
---|
487 | (mapcar #'(lambda (x) (normalize-poly x ring)) plist))
|
---|
488 |
|
---|
489 | (defun reduction (P pred ring)
|
---|
490 | "Reduce a list of polynomials P, so that non of the terms in any of the
|
---|
491 | polynomials is divisible by a leading monomial of another polynomial.
|
---|
492 | Return the reduced list."
|
---|
493 | (do ((Q P)
|
---|
494 | (found t))
|
---|
495 | ((not found)
|
---|
496 | (mapcar #'(lambda (x) (grobner-primitive-part x ring)) Q))
|
---|
497 | ;;Find p in Q such that p is reducible mod Q\{p}
|
---|
498 | (setf found nil)
|
---|
499 | (dolist (x Q)
|
---|
500 | (let ((Q1 (remove x Q)))
|
---|
501 | (multiple-value-bind (h div-count)
|
---|
502 | (normal-form x Q1
|
---|
503 | pred
|
---|
504 | nil ;not a top reduction
|
---|
505 | ring)
|
---|
506 | (unless (zerop div-count)
|
---|
507 | (setf found t Q Q1)
|
---|
508 | (when h
|
---|
509 | (setf Q (nconc Q1 (list h))))
|
---|
510 | (return)))))))
|
---|
511 |
|
---|
512 |
|
---|
513 | (defun reduced-grobner (F &optional (pred #'lex>)
|
---|
514 | (start 0)
|
---|
515 | (top-reduction-only nil)
|
---|
516 | (ring *coefficient-ring*))
|
---|
517 | "Return the reduced Grobner basis of the ideal generated by a polynomial
|
---|
518 | list F. This combines calls to two functions: GROBNER and REDUCTION.
|
---|
519 | The parameters have the same meaning as in GROBNER or BUCHBERGER."
|
---|
520 | (reduction (grobner F pred start top-reduction-only ring) pred ring))
|
---|
521 |
|
---|
522 | #+kcl
|
---|
523 | (defvar *vars* nil "Keep track of variables for tracing SPOLY.")
|
---|
524 |
|
---|
525 | ;; Does the monomial P depend on variable K
|
---|
526 | (defun monom-depends-p (m k)
|
---|
527 | "Return T if the monomial M depends on variable number K."
|
---|
528 | (plusp (elt m k)))
|
---|
529 |
|
---|
530 | ;; Does the term depend on variable K
|
---|
531 | (defun term-depends-p (term k)
|
---|
532 | "Return T if the term TERM depends on variable number K."
|
---|
533 | (monom-depends-p (car term) k))
|
---|
534 |
|
---|
535 | ;; Does the polynomial P depend on variable K
|
---|
536 | (defun poly-depends-p (p k)
|
---|
537 | "Return T if the term polynomial P depends on variable number K."
|
---|
538 | (some #'(lambda (term) (term-depends-p term k)) p))
|
---|
539 |
|
---|
540 | (defun ring-intersection (plist k &key (key #'identity))
|
---|
541 | "This function assumes that polynomial list PLIST is a Grobner basis
|
---|
542 | and it calculates the intersection with the ring R[x[k+1],...,x[n]], i.e.
|
---|
543 | it discards polynomials which depend on variables x[0], x[1], ..., x[k]."
|
---|
544 | (dotimes (i k plist)
|
---|
545 | (setf plist
|
---|
546 | (remove-if #'(lambda (p)
|
---|
547 | (poly-depends-p p (funcall key i)))
|
---|
548 | plist))))
|
---|
549 |
|
---|
550 | (defun elimination-ideal (flist k
|
---|
551 | &key
|
---|
552 | (primary-order #'lex>)
|
---|
553 | (secondary-order #'lex>)
|
---|
554 | (start 0)
|
---|
555 | (order (elimination-order
|
---|
556 | k
|
---|
557 | :primary-order primary-order
|
---|
558 | :secondary-order secondary-order
|
---|
559 | ))
|
---|
560 | (top-reduction-only nil)
|
---|
561 | (ring *coefficient-ring*))
|
---|
562 | "Returns the K-th elimination ideal of the ideal generated by polynomial list FLIST.
|
---|
563 | Thus, a Grobner basis of the ideal generated by FLIST is calculated and
|
---|
564 | polynomials depending on variables 0-K are discarded. The monomial order in the
|
---|
565 | calculation is an elimination order formed by combining two orders: PRIMARY-ORDER
|
---|
566 | used on variables 0-K and SECONDARY-ORDER used on variables starting from variable K+1.
|
---|
567 | Thus, if both orders are set to the lexicographic order #'LEX> then the resulting
|
---|
568 | order is simply equivalent to #'LEX>. But one could use arbitrary two orders in this
|
---|
569 | function, for instance, two copies of #'GREVLEX> (see ORDER package). When doing so,
|
---|
570 | the caller must ensure that the same order has been used to sort the terms of
|
---|
571 | the polynomials FLIST."
|
---|
572 | (ring-intersection
|
---|
573 | (reduced-grobner flist order start top-reduction-only ring)
|
---|
574 | k))
|
---|
575 |
|
---|
576 |
|
---|
577 | (defun ideal-intersection (F G pred top-reduction-only ring)
|
---|
578 | "Return the Grobner basis of the intersection of the ideals
|
---|
579 | generated by the polynomial lists F and G. PRED is an admissible
|
---|
580 | monomial order with respect to which the terms of F and G have been
|
---|
581 | sorted. This order is going to be used in the Grobner basis
|
---|
582 | calculation. If TOP-REDUCTION-ONLY is not NIL, internally the Grobner
|
---|
583 | basis algorithm will perform top reduction only. The RING parameter,
|
---|
584 | as usual, should be set to a structure defined in the package
|
---|
585 | COEFFICIENT-RING, and it will be used to perform all operations on
|
---|
586 | coefficients."
|
---|
587 | (mapcar
|
---|
588 | #'poly-contract
|
---|
589 | (ring-intersection
|
---|
590 | (reduced-grobner
|
---|
591 | (append
|
---|
592 | (mapcar #'(lambda (p) (poly-extend p (list 1))) F)
|
---|
593 | (mapcar #'(lambda (p)
|
---|
594 | (append (poly-extend (minus-poly p ring) (list 1))
|
---|
595 | (poly-extend p)))
|
---|
596 | G))
|
---|
597 | (elimination-order-1 pred)
|
---|
598 | 0
|
---|
599 | top-reduction-only
|
---|
600 | ring)
|
---|
601 | 1)))
|
---|
602 |
|
---|
603 |
|
---|
604 | (defun poly-contract (f &optional (k 1))
|
---|
605 | "Return a polynomial obtained from polynomial F by dropping the
|
---|
606 | first K variables, if F does not depend on them. Note that this
|
---|
607 | operation makes the polynomial incompatible for certain arithmetical
|
---|
608 | operations with the original polynomial F. Calling this function on a polynomial F
|
---|
609 | which does depend on variables 0-K will result in error."
|
---|
610 | (when (some #'(lambda (term) (find-if-not #'zerop (car term) :end k)) f)
|
---|
611 | (error "Contracting a polynomial which depends on contracted variables"))
|
---|
612 | (mapcar #'(lambda (term) (cons (nthcdr k (car term)) (cdr term))) f))
|
---|
613 |
|
---|
614 |
|
---|
615 | (defun poly-lcm (f g &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
616 | "Return LCM (least common multiple) of two polynomials F and G.
|
---|
617 | The polynomials must be ordered according to monomial order PRED
|
---|
618 | and their coefficients must be compatible with the RING structure
|
---|
619 | defined in the COEFFICIENT-RING package."
|
---|
620 | (cond
|
---|
621 | ((endp f) nil)
|
---|
622 | ((endp g) nil)
|
---|
623 | ((and (endp (cdr f)) (endp (cdr g)))
|
---|
624 | (list (cons (monom-lcm (caar f) (caar g))
|
---|
625 | (lcm (cdar f) (cdar g)))))
|
---|
626 | (t
|
---|
627 | (multiple-value-bind (f f-cont)
|
---|
628 | (grobner-primitive-part f ring)
|
---|
629 | (multiple-value-bind (g g-cont)
|
---|
630 | (grobner-primitive-part g ring)
|
---|
631 | (scalar-times-poly
|
---|
632 | (funcall (ring-lcm ring) f-cont g-cont)
|
---|
633 | (grobner-primitive-part (car (ideal-intersection (list f) (list g) pred nil ring))
|
---|
634 | ring)
|
---|
635 | ring))))))
|
---|
636 |
|
---|
637 | ;; GCD of two polynomials, using Grobner bases
|
---|
638 | (defun grobner-gcd (f g &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
639 | "Return GCD (greatest common divisor) of two polynomials F and G.
|
---|
640 | The polynomials must be ordered according to monomial order PRED
|
---|
641 | and their coefficients must be compatible with the RING structure
|
---|
642 | defined in the COEFFICIENT-RING package."
|
---|
643 | (poly-exact-divide (poly* f g pred ring)
|
---|
644 | (poly-lcm f g pred ring)
|
---|
645 | pred ring))
|
---|
646 |
|
---|
647 | ;; Do two Grobner bases yield the same ideal?
|
---|
648 | (defun grobner-equal (g1 g2 &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
649 | "Returns T if two lists of polynomials G1 and G2, assumed to be Grobner bases,
|
---|
650 | generate the same ideal, and NIL otherwise."
|
---|
651 | (and (grobner-subsetp g1 g2 pred ring)
|
---|
652 | (grobner-subsetp g2 g1 pred ring)))
|
---|
653 |
|
---|
654 | (defun grobner-subsetp (g1 g2 &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
655 | "Returns T if a list of polynomials G1 generates
|
---|
656 | an ideal contained in the ideal generated by a polynomial list G2,
|
---|
657 | both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise."
|
---|
658 | (every #'(lambda (p) (grobner-member p g2 pred ring)) g1))
|
---|
659 |
|
---|
660 | (defun grobner-member (p g &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
661 | "Returns T if a polynomial P belongs to the ideal generated by the
|
---|
662 | polynomial list G, which is assumed to be a Grobner basis. Returns NIL otherwise."
|
---|
663 | (endp (normal-form (copy-tree p) g pred nil ring)))
|
---|
664 |
|
---|
665 | (defun ideal-equal (f1 f2 pred top-reduction-only ring)
|
---|
666 | "Returns T if two ideals generated by polynomial lists F1 and F2 are identical.
|
---|
667 | Returns NIL otherwise."
|
---|
668 | (grobner-equal (grobner f1 pred 0 top-reduction-only ring)
|
---|
669 | (grobner f2 pred 0 top-reduction-only ring)
|
---|
670 | pred ring))
|
---|
671 |
|
---|
672 | (defun ideal-subsetp (f1 f2 &optional (pred #'lex>) (ring *coefficient-ring*))
|
---|
673 | "Returns T if the ideal spanned by the polynomial list F1 is contained
|
---|
674 | in the ideal spanned by the polynomial list F2. Returns NIL otherwise."
|
---|
675 | (grobner-subsetp
|
---|
676 | (grobner f1 pred 0 nil ring)
|
---|
677 | (grobner f2 pred 0 nil ring)
|
---|
678 | pred ring))
|
---|
679 |
|
---|
680 | (defun ideal-member (p plist pred top-reduction-only ring)
|
---|
681 | "Returns T if the polynomial P belongs to the ideal spanned by the polynomial
|
---|
682 | list PLIST, and NIL otherwise."
|
---|
683 | (endp (normal-form (copy-tree p) (grobner plist pred 0 top-reduction-only ring)
|
---|
684 | pred nil ring)))
|
---|
685 |
|
---|
686 | ;; Calculate F : p^inf
|
---|
687 | (defun ideal-saturation-1 (F p pred start top-reduction-only ring
|
---|
688 | &aux (pred (elimination-order-1 pred)))
|
---|
689 | "Returns the reduced Grobner basis of the saturation of the ideal
|
---|
690 | generated by a polynomial list F in the ideal generated by a single
|
---|
691 | polynomial P. The saturation ideal is defined as the set of
|
---|
692 | polynomials H such for some natural number n (* (EXPT P N) H) is in the ideal
|
---|
693 | F. Geometrically, over an algebraically closed field, this is the set
|
---|
694 | of polynomials in the ideal generated by F which do not identically
|
---|
695 | vanish on the variety of P."
|
---|
696 | (mapcar
|
---|
697 | #'poly-contract
|
---|
698 | (ring-intersection
|
---|
699 | (reduced-grobner
|
---|
700 | (saturation-extension-1 F p ring)
|
---|
701 | pred
|
---|
702 | start top-reduction-only ring)
|
---|
703 | 1)))
|
---|
704 |
|
---|
705 |
|
---|
706 | (defun add-variables (plist n)
|
---|
707 | "Add N new variables, adn the first N variables, to every polynomial in polynomial list PLIST.
|
---|
708 | The resulting polynomial will not depend on these N variables."
|
---|
709 | (mapcar
|
---|
710 | #'(lambda (g)
|
---|
711 | (mapcar #'(lambda (q) (cons (append (make-list n :initial-element 0)
|
---|
712 | (car q))
|
---|
713 | (cdr q))) g))
|
---|
714 | plist))
|
---|
715 |
|
---|
716 | ;; Calculate <u1*p1,u2*p2,...,uk*pk>
|
---|
717 | (defun extend-polynomials (plist &aux (k (length plist)))
|
---|
718 | "Returns polynomial list {U1*P1', U2*P2', ... , UK*PK'}
|
---|
719 | where Ui are new variables and PLIST={P1, P2, ... , PK} is a polynomial
|
---|
720 | list. PI' is obtained from PI by adding new variables U1, U2, ..., UN
|
---|
721 | UK as the first K variables. Thus, the resulting list is consists of polynomials whose
|
---|
722 | terms are ordered by the lexicographic order on variables UI, with
|
---|
723 | ties resolved by the monomial order which was used to order the terms
|
---|
724 | of the polynomials in PLIST. We note that the monomial order does not
|
---|
725 | explicitly participate in this calculation, and neither do the
|
---|
726 | variable names."
|
---|
727 | (labels ((incf-power (g i)
|
---|
728 | (dolist (x g g)
|
---|
729 | (incf (nth i (car x))))))
|
---|
730 | (setf plist (add-variables plist k))
|
---|
731 | (dotimes (i k plist)
|
---|
732 | (setf (nth i plist) (incf-power (nth i plist) i)))))
|
---|
733 |
|
---|
734 | ;; Calculate <F, u1*p1-1,u2*p2-1,...,uk*pk-1>
|
---|
735 | (defun saturation-extension (F plist ring &aux (k (length plist))
|
---|
736 | (d (+ k (length (caaar plist)))))
|
---|
737 | "Returns F' union {U1*P1-1,U2*P2-1,...,UK*PK-1} where Ui are new variables
|
---|
738 | and F' is a polynomial list obtained from F by adding variables U1, U2, ..., Uk
|
---|
739 | as the first K variables to each polynomial in F. Thus, the resulting list is
|
---|
740 | consists of polynomials whose terms are ordered by the lexicographic order on
|
---|
741 | variables Ui, with ties resolved by the monomial order which was used to order the
|
---|
742 | terms of the polynomials in F. We note that the monomial order does not explicitly
|
---|
743 | participate in this calculation, and neither do the variable names."
|
---|
744 | (setf F (add-variables F k)
|
---|
745 | plist (mapcar #'(lambda (x)
|
---|
746 | (nconc x (list (cons (make-list d :initial-element 0)
|
---|
747 | (funcall (ring-- ring)
|
---|
748 | (ring-unit ring))))))
|
---|
749 | (extend-polynomials plist)))
|
---|
750 | (append F plist))
|
---|
751 |
|
---|
752 | ;; Calculate <F, u1*p1+u2*p2+...+uk*pk-1>
|
---|
753 | (defun polysaturation-extension (F plist ring &aux (k (length plist))
|
---|
754 | (d (+ k (length (caaar plist)))))
|
---|
755 | "Returns F' union {U1*P1+U2*P2+UK*PK-1} where Ui are new variables
|
---|
756 | and F' is a polynomial list obtained from F by adding variables U1, U2, ..., Uk
|
---|
757 | as the first K variables to each polynomial in F. Thus, the resulting list is
|
---|
758 | consists of polynomials whose terms are ordered by the lexicographic order on
|
---|
759 | variables Ui, with ties resolved by the monomial order which was used to order the
|
---|
760 | terms of the polynomials in F. We note that the monomial order does not explicitly
|
---|
761 | participate in this calculation, and neither do the variable names."
|
---|
762 | (setf F (add-variables F k)
|
---|
763 | plist (apply #'append (extend-polynomials plist))
|
---|
764 | (cdr (last plist)) (list (cons (make-list d :initial-element 0)
|
---|
765 | (funcall (ring-- ring)
|
---|
766 | (ring-unit ring)))))
|
---|
767 | (append F (list plist)))
|
---|
768 |
|
---|
769 | (defun saturation-extension-1 (F p ring)
|
---|
770 | "Return the list F' union {U*P-1} where U is a new variable,
|
---|
771 | where F' is obtained from the polynomial list F by adding U as the first variable.
|
---|
772 | The variable U will be added as the first variable, so that it can be easily
|
---|
773 | eliminated."
|
---|
774 | (saturation-extension F (list p) ring))
|
---|
775 |
|
---|
776 | ;; Calculate F : p1^inf : p2^inf : ... : ps^inf
|
---|
777 | (defun ideal-polysaturation-1 (F plist pred start top-reduction-only ring)
|
---|
778 | "Returns the reduced Grobner basis of the ideal obtained by a
|
---|
779 | sequence of successive saturations in the polynomials
|
---|
780 | of the polynomial list PLIST of the ideal generated by the
|
---|
781 | polynomial list F."
|
---|
782 | (cond
|
---|
783 | ((endp plist) (reduced-grobner F pred start top-reduction-only ring))
|
---|
784 | (t (let ((G (ideal-saturation-1 F (car plist) pred start top-reduction-only ring)))
|
---|
785 | (ideal-polysaturation-1 G (rest plist) pred (length G) top-reduction-only ring)))))
|
---|
786 |
|
---|
787 | (defun ideal-saturation (F G pred start top-reduction-only ring
|
---|
788 | &aux (k (length G))
|
---|
789 | (pred (elimination-order k
|
---|
790 | :primary-order #'lex>
|
---|
791 | :secondary-order pred)))
|
---|
792 | "Returns the reduced Grobner basis of the saturation of the ideal
|
---|
793 | generated by a polynomial list F in the ideal generated a polynomial
|
---|
794 | list G The saturation ideal is defined as the set of polynomials H
|
---|
795 | such for some natural number n and some P in the ideal generated by G
|
---|
796 | the polynomial P**N * H is in the ideal spanned by F. Geometrically,
|
---|
797 | over an algebraically closed field, this is the set of polynomials in
|
---|
798 | the ideal generated by F which do not identically vanish on the
|
---|
799 | variety of G."
|
---|
800 | (mapcar
|
---|
801 | #'(lambda (q) (poly-contract q k))
|
---|
802 | (ring-intersection
|
---|
803 | (reduced-grobner (polysaturation-extension F G ring)
|
---|
804 | pred start
|
---|
805 | top-reduction-only ring)
|
---|
806 | k)))
|
---|
807 |
|
---|
808 | (defun ideal-polysaturation (F ideal-list pred start top-reduction-only ring)
|
---|
809 | "Returns the reduced Grobner basis of the ideal obtained by a
|
---|
810 | successive applications of IDEAL-SATURATIONS to F and
|
---|
811 | lists of polynomials in the list IDEAL-LIST."
|
---|
812 | (cond
|
---|
813 | ((endp ideal-list) F)
|
---|
814 | (t (let ((H (ideal-saturation F (car ideal-list) pred start top-reduction-only ring)))
|
---|
815 | (ideal-polysaturation H (rest ideal-list) pred (length H) top-reduction-only ring)))))
|
---|
816 |
|
---|
817 | (defun buchberger-criterion (G pred ring)
|
---|
818 | "Returns T if G is a Grobner basis, by using the Buchberger
|
---|
819 | criterion: for every two polynomials h1 and h2 in G the S-polynomial
|
---|
820 | S(h1,h2) reduces to 0 modulo G."
|
---|
821 | (every
|
---|
822 | #'endp
|
---|
823 | (makelist (normal-form (spoly (elt G i) (elt G j) pred ring) G pred nil ring)
|
---|
824 | (i 0 (- (length G) 2))
|
---|
825 | (j (1+ i) (1- (length G))))))
|
---|
826 |
|
---|
827 |
|
---|
828 | (defun grobner-test (G F pred ring)
|
---|
829 | "Test whether G is a Grobner basis and F is contained in G. Return T
|
---|
830 | upon success and NIL otherwise."
|
---|
831 | #+debug(debug-cgb "~&GROBNER CHECK: ")
|
---|
832 | (let ((*grobner-debug* nil)
|
---|
833 | (stat1 (buchberger-criterion G pred ring))
|
---|
834 | (stat2
|
---|
835 | (every #'endp
|
---|
836 | (makelist (normal-form (copy-tree (elt F i)) G pred nil ring)
|
---|
837 | (i 0 (1- (length F)))))))
|
---|
838 | (unless stat1 (error "~&Buchberger criterion failed."))
|
---|
839 | (unless stat2
|
---|
840 | (error "~&Original polys not in ideal spanned by Grobner.")))
|
---|
841 | #+debug(debug-cgb "~&GROBNER CHECK END")
|
---|
842 | T)
|
---|
843 |
|
---|
844 |
|
---|
845 | (defun minimization (P pred)
|
---|
846 | "Returns a sublist of the polynomial list P spanning the same
|
---|
847 | monomial ideal as P but minimal, i.e. no leading monomial
|
---|
848 | of a polynomial in the sublist divides the leading monomial
|
---|
849 | of another polynomial."
|
---|
850 | (declare (ignore pred))
|
---|
851 | (do ((Q P)
|
---|
852 | (found t))
|
---|
853 | ((not found) Q)
|
---|
854 | ;;Find p in Q such that lm(p) is in LM(Q\{p})
|
---|
855 | (setf found nil
|
---|
856 | Q (dolist (x Q Q)
|
---|
857 | (let ((Q1 (remove x Q)))
|
---|
858 | (when (member-if #'(lambda (p) (monom-divides-p (lm x) (lm p))) Q1)
|
---|
859 | (setf found t)
|
---|
860 | (return Q1)))))))
|
---|
861 |
|
---|
862 |
|
---|
863 | (defun add-minimized (f Gred pred)
|
---|
864 | "Adds a polynomial f to GRED, reduced Grobner basis, preserving the
|
---|
865 | property described in the documentation for MINIMIZATION."
|
---|
866 | (if (member-if #'(lambda (p) (monom-divides-p (lm p) (lm f))) Gred)
|
---|
867 | Gred
|
---|
868 | (merge 'list (list f)
|
---|
869 | (delete-if #'(lambda (p) (monom-divides-p (lm f) (lm p))) Gred)
|
---|
870 | pred :key #'lm)))
|
---|
871 |
|
---|
872 | (defun colon-ideal (F G pred top-reduction-only ring)
|
---|
873 | "Returns the reduced Grobner basis of the colon ideal Id(F):Id(G), where
|
---|
874 | F and G are two lists of polynomials. The colon ideal I:J is defined as the
|
---|
875 | set of polynomials H such that there is a polynomial W in J for which
|
---|
876 | W*H belongs to I."
|
---|
877 | (cond
|
---|
878 | ((endp G)
|
---|
879 | (if (every #'endp F) nil
|
---|
880 | (list (list
|
---|
881 | (cons
|
---|
882 | (make-list (length (caar (find-if-not #'endp F)))
|
---|
883 | :initial-element 0)
|
---|
884 | 1)))))
|
---|
885 | ((endp (cdr G))
|
---|
886 | (colon-ideal-1 F (car G) pred top-reduction-only ring))
|
---|
887 | (t
|
---|
888 | (ideal-intersection (colon-ideal-1 F (car G) pred top-reduction-only ring)
|
---|
889 | (colon-ideal F (rest G) pred top-reduction-only ring)
|
---|
890 | pred
|
---|
891 | top-reduction-only
|
---|
892 | ring))))
|
---|
893 |
|
---|
894 | (defun colon-ideal-1 (F g pred top-reduction-only ring)
|
---|
895 | "Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}), where
|
---|
896 | F is a list of polynomials and G is a polynomial."
|
---|
897 | (mapcar #'(lambda (x) (poly-exact-divide x g pred ring))
|
---|
898 | (ideal-intersection F (list g) pred top-reduction-only ring)))
|
---|
899 |
|
---|
900 |
|
---|
901 | (defun pseudo-divide (f fl pred ring)
|
---|
902 | "Pseudo-divide a polynomial F by the list of polynomials FL. Return
|
---|
903 | multiple values. The first value is a list of quotients A.
|
---|
904 | The second value is the remainder R. The third value is an integer
|
---|
905 | count of the number of reductions performed. Finally, the fourth
|
---|
906 | argument is a scalar coefficient C, such that C*F can be divided by FL
|
---|
907 | within the ring of coefficients, which is not necessarily a field.
|
---|
908 | The resulting objects satisfy the equation:
|
---|
909 | C*F= sum A[i]*FL[i] + R
|
---|
910 | "
|
---|
911 | (declare (optimize (speed 3) (safety 0)))
|
---|
912 | (do (r
|
---|
913 | (c (ring-unit ring))
|
---|
914 | (a (make-list (length fl)))
|
---|
915 | (division-count 0)
|
---|
916 | (p f))
|
---|
917 | ((endp p)
|
---|
918 | #+debug(debug-cgb "~&~3T~d reduction~:p" division-count)
|
---|
919 | #+debug(when (endp r) (debug-cgb " ---> 0"))
|
---|
920 | (values a (nreverse r) division-count c))
|
---|
921 | (declare (fixnum division-count) (integer c))
|
---|
922 | (do ((fl fl (rest fl)) ;scan list of divisors
|
---|
923 | (b a (rest b)))
|
---|
924 | ((cond
|
---|
925 | ((endp fl) ;no division occurred
|
---|
926 | (setf r (cons (lt p) r) ;move lt(p) to remainder
|
---|
927 | p (rest p)) ;remove lt(p) from p
|
---|
928 | t)
|
---|
929 | ((monom-divides-p (lm (car fl)) (lm p)) ;division occurred
|
---|
930 | (incf division-count)
|
---|
931 | (let* ((gcd (funcall (ring-gcd ring) (lc (car fl)) (lc p)))
|
---|
932 | (c1 (funcall (ring-/ ring) (lc (car fl)) gcd))
|
---|
933 | (c2 (funcall (ring-/ ring) (lc p) gcd))
|
---|
934 | (quot (cons (monom/ (lm p) (lm (car fl))) c2)))
|
---|
935 | ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
|
---|
936 | (mapl #'(lambda (x)
|
---|
937 | (setf (car x) (scalar-times-poly c1 (car x) ring)))
|
---|
938 | a)
|
---|
939 | (setf p (scalar-times-poly c1 (rest p) ring) ;drop first term
|
---|
940 | r (scalar-times-poly c1 r ring)
|
---|
941 | c (funcall (ring-* ring) c c1)
|
---|
942 | p (poly-op p quot (rest (car fl)) pred ring)
|
---|
943 | (car b) (nconc (car b) (list quot))))
|
---|
944 | t))))))
|
---|
945 |
|
---|
946 | ;;----------------------------------------------------------------
|
---|
947 | ;; An implementation of the algorithm of Gebauer and Moeller,
|
---|
948 | ;; as described in the book of Becker-Weispfenning, p. 232
|
---|
949 | ;;----------------------------------------------------------------
|
---|
950 | (defun gebauer-moeller (F pred start top-reduction-only ring
|
---|
951 | &aux B G F1)
|
---|
952 | "Compute Grobner basis by using the algorithm of Gebauer and Moeller.
|
---|
953 | This algorithm is described as BUCHBERGERNEW2 in the book by
|
---|
954 | Becker-Weispfenning entitled ``Grobner Bases''"
|
---|
955 | ;;cut startup costs if (length F)<=1
|
---|
956 | (cond
|
---|
957 | ((endp F) (return-from gebauer-moeller nil))
|
---|
958 | ((endp (cdr F))
|
---|
959 | (return-from gebauer-moeller (list (grobner-primitive-part (car F) ring)))))
|
---|
960 | #+debug (debug-cgb "~&GROBNER BASIS - GEBAUER MOELLER ALGORITHM")
|
---|
961 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
|
---|
962 | #+grobner-check (when (plusp start)
|
---|
963 | (grobner-test (subseq F 0 start) (subseq F 0 start)
|
---|
964 | pred ring))
|
---|
965 | (setf B nil
|
---|
966 | G (subseq F 0 start)
|
---|
967 | F1 (subseq F start))
|
---|
968 | (do () ((endp F1))
|
---|
969 | (multiple-value-setq (G B)
|
---|
970 | (update G B (grobner-primitive-part (pop F1) ring) pred ring)))
|
---|
971 | (do () ((endp B))
|
---|
972 | (let* ((pair (pop B))
|
---|
973 | (g1 (first pair))
|
---|
974 | (g2 (second pair))
|
---|
975 | (h (normal-form (spoly g1 g2 pred ring)
|
---|
976 | G pred top-reduction-only ring)))
|
---|
977 | (unless (poly-zerop h)
|
---|
978 | (setf h (grobner-primitive-part h ring))
|
---|
979 | (multiple-value-setq (G B)
|
---|
980 | (update G B h pred ring))
|
---|
981 | #+debug (debug-cgb "~&Polynomials: ~d; Pairs left: ~d~%"
|
---|
982 | (length G) (length B))
|
---|
983 | )))
|
---|
984 | #+grobner-check(grobner-test G F pred ring)
|
---|
985 | #+debug(debug-cgb "~&GROBNER END")
|
---|
986 | G)
|
---|
987 |
|
---|
988 | (defun update (G B h pred ring &aux C D E B-new G-new pair)
|
---|
989 | "An implementation of the auxillary UPDATE algorithm used by the
|
---|
990 | Gebauer-Moeller algorithm. G is a list of polynomials, B is a list of
|
---|
991 | critical pairs and H is a new polynomial which possibly will be added
|
---|
992 | to G. The naming conventions used are very close to the one used in
|
---|
993 | the book of Becker-Weispfenning."
|
---|
994 | (setf C G D nil)
|
---|
995 | (do (g1) ((endp C))
|
---|
996 | (setf g1 (pop C))
|
---|
997 | (when (or (monom-rel-prime (lm h) (lm g1))
|
---|
998 | (and
|
---|
999 | (not (some
|
---|
1000 | #'(lambda (g2) (monom-divides-p (monom-lcm (lm h) (lm g2))
|
---|
1001 | (monom-lcm (lm h) (lm g1))))
|
---|
1002 | C))
|
---|
1003 | (not (some
|
---|
1004 | #'(lambda (g2) (monom-divides-p (monom-lcm (lm h) (lm g2))
|
---|
1005 | (monom-lcm (lm h) (lm g1))))
|
---|
1006 | D))))
|
---|
1007 | (push g1 D)))
|
---|
1008 | (setf E nil)
|
---|
1009 | (do (g1) ((endp D))
|
---|
1010 | (setf g1 (pop D))
|
---|
1011 | (unless (monom-rel-prime (lm h) (lm g1))
|
---|
1012 | (push g1 E)))
|
---|
1013 | (setf B-new nil)
|
---|
1014 | (do (g1 g2) ((endp B))
|
---|
1015 | (setf pair (pop B)
|
---|
1016 | g1 (first pair)
|
---|
1017 | g2 (second pair))
|
---|
1018 | (when (or (not (monom-divides-p (lm h) (monom-lcm (lm g1) (lm g2))))
|
---|
1019 | (equal (monom-lcm (lm g1) (lm h))
|
---|
1020 | (monom-lcm (lm g1) (lm g2)))
|
---|
1021 | (equal (monom-lcm (lm h) (lm g2))
|
---|
1022 | (monom-lcm (lm g1) (lm g2))))
|
---|
1023 | (setf B-new (funcall *gebauer-moeller-merge-pairs* B-new (list (list g1 g2)) pred ring))))
|
---|
1024 | (dolist (g3 E)
|
---|
1025 | (setf B-new (funcall *gebauer-moeller-merge-pairs* B-new (list (list h g3)) pred ring)))
|
---|
1026 | (setf G-new nil)
|
---|
1027 | (do (g1) ((endp G))
|
---|
1028 | (setf g1 (pop G))
|
---|
1029 | (unless (monom-divides-p (lm h) (lm g1))
|
---|
1030 | (push g1 G-new)))
|
---|
1031 | (push h G-new)
|
---|
1032 | (values G-new B-new))
|
---|
1033 |
|
---|
1034 | (defun gebauer-moeller-merge-pairs-use-mock-spoly (B C pred ring)
|
---|
1035 | "The merging strategy used by the Gebauer-Moeller algorithm, based
|
---|
1036 | on MOCK-SPOLY; see the documentation of
|
---|
1037 | BUCHBERGER-MERGE-PAIRS-USE-MOCK-SPOLY."
|
---|
1038 | (declare (ignore ring))
|
---|
1039 | ;; Delete pairs consisting of two single monomials
|
---|
1040 | (setf C (delete-if
|
---|
1041 | #'(lambda (p) (and (endp (cdr (first p)))
|
---|
1042 | (endp (cdr (second p)))))
|
---|
1043 | C))
|
---|
1044 | (labels ((pair-key (p) (mock-spoly (first p) (second p) pred))
|
---|
1045 | (pair-order (x y) (funcall pred y x)))
|
---|
1046 | (merge 'list B C #'pair-order :key #'pair-key)))
|
---|
1047 |
|
---|
1048 | (defun gebauer-moeller-merge-pairs-smallest-lcm (B C pred ring)
|
---|
1049 | "The merging strategy based on the smallest-lcm (normal strategy); see
|
---|
1050 | the documentation of BUCHBERGER-MERGE-PAIRS-SMALLEST-LCM."
|
---|
1051 | (declare (ignore ring))
|
---|
1052 | ;; Delete pairs consisting of two single monomials
|
---|
1053 | (labels ((pair-key (p) (monom-lcm (lm (first p))
|
---|
1054 | (lm (second p))))
|
---|
1055 | (pair-order (x y) (funcall pred y x)))
|
---|
1056 | (merge 'list B C #'pair-order :key #'pair-key)))
|
---|
1057 |
|
---|
1058 |
|
---|
1059 | (defun gebauer-moeller-merge-pairs-use-smallest-degree (B C pred ring)
|
---|
1060 | "The merging strategy based on the smallest-lcm (normal strategy); see
|
---|
1061 | the documentation of BUCHBERGER-MERGE-PAIRS-USE-SMALLEST-DEGREE."
|
---|
1062 | (declare (ignore ring))
|
---|
1063 | (declare (ignore pred))
|
---|
1064 | (labels ((pair-key (p) (total-degree
|
---|
1065 | (monom-lcm (lm (first p))
|
---|
1066 | (lm (second p)))))
|
---|
1067 | (pair-order (x y) (funcall #'< x y)))
|
---|
1068 | (merge 'list B C #'pair-order :key #'pair-key)))
|
---|
1069 |
|
---|
1070 |
|
---|
1071 | (defun gebauer-moeller-merge-pairs-use-smallest-length (B C pred ring)
|
---|
1072 | (declare (ignore ring))
|
---|
1073 | "The merging strategy based on the smallest-lcm (normal strategy); see
|
---|
1074 | the documentation of BUCHBERGER-MERGE-PAIRS-USE-SMALLEST-LENGTH."
|
---|
1075 | (declare (ignore pred))
|
---|
1076 | (labels ((pair-key (p) (+
|
---|
1077 | (length (first p))
|
---|
1078 | (length (second p))))
|
---|
1079 | (pair-order (x y) (funcall #'< x y)))
|
---|
1080 | (if B
|
---|
1081 | (merge 'list B C #'pair-order :key #'pair-key)
|
---|
1082 | (sort C #'pair-order :key #'pair-key))))
|
---|
1083 |
|
---|
1084 |
|
---|
1085 | (defun gebauer-moeller-merge-pairs-use-smallest-coefficient-length (B C pred ring)
|
---|
1086 | "The merging strategy based on the smallest-lcm (normal strategy); see
|
---|
1087 | the documentation of BUCHBERGER-MERGE-PAIRS-USE-SMALLEST-COEFFICIENT-LENGTH."
|
---|
1088 | (declare (ignore pred))
|
---|
1089 | (labels ((coefficient-length (poly)
|
---|
1090 | (apply #'+ (mapcar (ring-length ring) (mapcar #'cdr poly))))
|
---|
1091 | (pair-key (p) (+
|
---|
1092 | (coefficient-length (first p))
|
---|
1093 | (coefficient-length (second p))))
|
---|
1094 | (pair-order (x y) (funcall #'< x y)))
|
---|
1095 | (merge 'list B C #'pair-order :key #'pair-key)))
|
---|
1096 |
|
---|
1097 | (defun gebauer-moeller-set-pair-heuristic
|
---|
1098 | (method
|
---|
1099 | &aux (strategy-fn
|
---|
1100 | (ecase method
|
---|
1101 | (:minimal-mock-spoly #'gebauer-moeller-merge-pairs-use-mock-spoly)
|
---|
1102 | (:minimal-lcm #'gebauer-moeller-merge-pairs-smallest-lcm)
|
---|
1103 | (:minimal-total-degree #'gebauer-moeller-merge-pairs-use-smallest-degree)
|
---|
1104 | (:minimal-length #'gebauer-moeller-merge-pairs-use-smallest-length)
|
---|
1105 | (:minimal-coefficient-length #'gebauer-moeller-merge-pairs-use-smallest-coefficient-length))))
|
---|
1106 | (setf *gebauer-moeller-merge-pairs* strategy-fn))
|
---|
1107 |
|
---|
1108 | ;;------------------------------------------------------------------------------------------------
|
---|
1109 | ;;
|
---|
1110 | ;; An implementation of the sugar strategy
|
---|
1111 | ;;
|
---|
1112 | ;;------------------------------------------------------------------------------------------------
|
---|
1113 |
|
---|
1114 | ;; Calculate sugar of the S-polynomial of f and g
|
---|
1115 | (defun spoly-sugar (f-with-sugar g-with-sugar ring)
|
---|
1116 | "Calculate the ``sugar'' of the S-polynomial of two polynomials with
|
---|
1117 | sugar. A polynomial with sugar is simply a pair (P . SUGAR) where
|
---|
1118 | SUGAR is an integer constant defined according to the following
|
---|
1119 | algorighm: the sugar of sum or difference of two polynomials with
|
---|
1120 | sugar is the MAX of the sugars of those two polynomials. The sugar of
|
---|
1121 | a product of a term and a polynomial is the sum of the degree of the
|
---|
1122 | term and the sugar of the polynomial. The idea is to accumulate sugar
|
---|
1123 | as we perform the arithmetic operations, and that polynomials with
|
---|
1124 | small (little) sugar should be given priority in the calculations.
|
---|
1125 | Thus, the ``sugar strategy'' of the critical pair selection is to
|
---|
1126 | select a pair with the smallest value of sugar of the resulting
|
---|
1127 | S-polynomial."
|
---|
1128 | (let* ((f (poly-with-sugar-poly f-with-sugar))
|
---|
1129 | (g (poly-with-sugar-poly g-with-sugar))
|
---|
1130 | (lcm (monom-lcm (lm f) (lm g)))
|
---|
1131 | (m1 (monom/ lcm (lm f)))
|
---|
1132 | (m2 (monom/ lcm (lm g))))
|
---|
1133 | (let* ((c (funcall (ring-lcm ring) (lc f) (lc g)))
|
---|
1134 | (cf (funcall (ring-/ ring) c (lc f)))
|
---|
1135 | (cg (funcall (ring-/ ring) c (lc g))))
|
---|
1136 | (max
|
---|
1137 | (+ (term-sugar (cons m1 cf) ring) (poly-with-sugar-sugar f-with-sugar))
|
---|
1138 | (+ (term-sugar (cons m2 cg) ring) (poly-with-sugar-sugar g-with-sugar))))))
|
---|
1139 |
|
---|
1140 | ;; Calculate the S-polynomial of f and g with sugar
|
---|
1141 | (defun spoly-with-sugar (f-with-sugar g-with-sugar pred ring)
|
---|
1142 | "The S-polynomials of two polynomials with SUGAR strategy."
|
---|
1143 | (let* ((f (poly-with-sugar-poly f-with-sugar))
|
---|
1144 | (g (poly-with-sugar-poly g-with-sugar))
|
---|
1145 | (lcm (monom-lcm (lm f) (lm g)))
|
---|
1146 | (m1 (monom/ lcm (lm f)))
|
---|
1147 | (m2 (monom/ lcm (lm g))))
|
---|
1148 | (let* ((c (funcall (ring-lcm ring) (lc f) (lc g)))
|
---|
1149 | (cf (funcall (ring-/ ring) c (lc f)))
|
---|
1150 | (cg (funcall (ring-/ ring) c (lc g))))
|
---|
1151 | (poly-with-sugar-
|
---|
1152 | (term-times-poly-with-sugar (cons m1 cf) (poly-with-sugar-tail f-with-sugar) ring)
|
---|
1153 | (term-times-poly-with-sugar (cons m2 cg) (poly-with-sugar-tail g-with-sugar) ring)
|
---|
1154 | pred
|
---|
1155 | ring))))
|
---|
1156 |
|
---|
1157 |
|
---|
1158 | (defun normal-form-with-sugar (f fl pred top-reduction-only ring)
|
---|
1159 | "Normal form of the polynomial with sugar F with respect to
|
---|
1160 | a list of polynomials with sugar FL. Assumes that FL is not empty.
|
---|
1161 | Parameters PRED, TOP-REDUCTION-ONLY and RING have the same meaning as in NORMAL-FORM.
|
---|
1162 | The parameter SUGAR-LIMIT should be set to a positive integer. If the sugar limit of
|
---|
1163 | the partial remainder exceeds SUGAR-LIMIT then the calculation is stopped and the partial
|
---|
1164 | remainder is returned, although it is not fully reduced with respect to FL."
|
---|
1165 | (declare (optimize (speed 3) (safety 0)))
|
---|
1166 | ;; Loop invariant: c*f=sum ai*fi+r+p
|
---|
1167 | (do ((r (cons nil 0))
|
---|
1168 | (c (ring-unit ring))
|
---|
1169 | (division-count 0)
|
---|
1170 | (p f))
|
---|
1171 | ((or (poly-with-sugar-zerop p)
|
---|
1172 | (and top-reduction-only (not (poly-with-sugar-zerop r))))
|
---|
1173 | #+debug
|
---|
1174 | (progn
|
---|
1175 | (debug-cgb "~&~3T~d reduction~:p" division-count)
|
---|
1176 | (cond
|
---|
1177 | ((poly-with-sugar-zerop r)
|
---|
1178 | (debug-cgb " ---> 0"))))
|
---|
1179 | (values (poly-with-sugar-append (poly-with-sugar-nreverse r) p)
|
---|
1180 | division-count c))
|
---|
1181 | (declare (fixnum division-count) (integer c))
|
---|
1182 | (let ((g (find (poly-with-sugar-lm p) fl
|
---|
1183 | :test #'monom-divisible-by-p
|
---|
1184 | :key #'poly-with-sugar-lm)))
|
---|
1185 | (cond
|
---|
1186 | (g ;division possible
|
---|
1187 | (incf division-count)
|
---|
1188 | (let* ((gcd (funcall (ring-gcd ring) (poly-with-sugar-lc g) (poly-with-sugar-lc p)))
|
---|
1189 | (c1 (funcall (ring-/ ring) (poly-with-sugar-lc g) gcd))
|
---|
1190 | (c2 (funcall (ring-/ ring) (poly-with-sugar-lc p) gcd))
|
---|
1191 | (m (monom/ (poly-with-sugar-lm p) (poly-with-sugar-lm g))))
|
---|
1192 | ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
|
---|
1193 | (setf r (scalar-times-poly-with-sugar c1 r ring)
|
---|
1194 | c (funcall (ring-* ring) c c1)
|
---|
1195 | p (scalar-times-poly-with-sugar c1 p ring)
|
---|
1196 | p (poly-with-sugar-op p (cons m c2) g pred ring))))
|
---|
1197 | (t ;no division possible
|
---|
1198 | ;;move lt(p) to remainder
|
---|
1199 | (setf r (cons (cons (poly-with-sugar-lt p) (poly-with-sugar-poly r))
|
---|
1200 | (poly-with-sugar-sugar r))
|
---|
1201 | p (poly-with-sugar-tail p)))))))
|
---|
1202 |
|
---|
1203 | ;; Parameter START is used when Grobner basis is calculated incrementally
|
---|
1204 | ;; and it signals that the elements of F from 0 to START-1 form a Grobner
|
---|
1205 | ;; basis already, so their pairs are not formed.
|
---|
1206 | (defun buchberger-with-sugar (F-no-sugar pred start top-reduction-only ring
|
---|
1207 | &aux (s (1- (length F-no-sugar))) B M F)
|
---|
1208 | "Returns a Grobner basis of an ideal generated by a list of ordinary
|
---|
1209 | polynomials F-NO-SUGAR. Adds initial sugar to the polynomials and
|
---|
1210 | performs the Buchberger algorithm with ``sugar strategy''. It returns
|
---|
1211 | an ordinary list of polynomials with no sugar. One of the most
|
---|
1212 | effective algorithms for Grobner basis calculation."
|
---|
1213 | (declare (fixnum s))
|
---|
1214 | (unless (plusp s) (return-from buchberger-with-sugar F-no-sugar)) ;cut startup costs
|
---|
1215 | #+debug (debug-cgb "~&GROBNER BASIS - BUCHBERGER WITH SUGER STRATEGY ALGORITHM")
|
---|
1216 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
|
---|
1217 | #+grobner-check (when (plusp start)
|
---|
1218 | (grobner-test (subseq F-no-sugar 0 start)
|
---|
1219 | (subseq F-no-sugar 0 start) pred ring))
|
---|
1220 | (setf F (mapcar #'(lambda (x) (poly-add-sugar x ring)) F-no-sugar)
|
---|
1221 | B (nconc (makelist (list i j) (i 0 (1- start)) (j start s))
|
---|
1222 | (makelist (list i j) (i start (1- s)) (j (1+ i) s)))
|
---|
1223 | M (make-hash-table :test #'equal))
|
---|
1224 | ;;Initialize treated pairs
|
---|
1225 | (dotimes (i (1- start))
|
---|
1226 | (do ((j (1+ i) (1+ j))) ((>= j start))
|
---|
1227 | (setf (gethash (list i j) M) t)))
|
---|
1228 | (setf B (buchberger-with-sugar-sort-pairs B F M pred ring))
|
---|
1229 | (do ((G F))
|
---|
1230 | ((endp B)
|
---|
1231 | (setf G (mapcar #'poly-with-sugar-poly G))
|
---|
1232 | ;;#+grobner-check(grobner-test G F-no-sugar pred ring)
|
---|
1233 | #+debug(debug-cgb "~&GROBNER END")
|
---|
1234 | G)
|
---|
1235 | (let ((pair (pop B)))
|
---|
1236 | (cond
|
---|
1237 | ((Criterion-1-with-sugar pair G))
|
---|
1238 | ((Criterion-2-with-sugar pair G M))
|
---|
1239 | (t
|
---|
1240 | (let ((SP (normal-form-with-sugar
|
---|
1241 | (cons (spoly (poly-with-sugar-poly (elt G (first pair)))
|
---|
1242 | (poly-with-sugar-poly (elt G (second pair))) pred ring)
|
---|
1243 | (third pair))
|
---|
1244 | G pred top-reduction-only ring)))
|
---|
1245 | (cond
|
---|
1246 | ((poly-with-sugar-zerop SP))
|
---|
1247 | (t
|
---|
1248 | (setf s (1+ s)
|
---|
1249 | (poly-with-sugar-poly SP) (grobner-primitive-part
|
---|
1250 | (poly-with-sugar-poly SP) ring)
|
---|
1251 | (cdr (last G)) (list SP)
|
---|
1252 | B (buchberger-with-sugar-merge-pairs B (makelist (list i s)
|
---|
1253 | (i 0 (1- s)))
|
---|
1254 | G M pred ring))
|
---|
1255 | #+debug (debug-cgb "~&[~d] Polynomials: ~d; Pairs left: ~d; Treated pairs: ~d"
|
---|
1256 | (third pair)
|
---|
1257 | (length G) (length B)
|
---|
1258 | (hash-table-count M)))))))
|
---|
1259 | (setf (cddr pair) nil
|
---|
1260 | (gethash pair M) t))))
|
---|
1261 |
|
---|
1262 |
|
---|
1263 | (defun buchberger-with-sugar-merge-pairs (B C G M pred ring)
|
---|
1264 | "Merges lists of critical pairs. It orders pairs according to
|
---|
1265 | increasing sugar, with ties broken by smaller MOCK-SPOLY value of the
|
---|
1266 | two polynomials. In this function B must already be sorted."
|
---|
1267 | ;;Remove pairs consisting of two monomials
|
---|
1268 | ;;These pairs produce zero S-polynomial
|
---|
1269 | (setf C (delete-if
|
---|
1270 | #'(lambda (p)
|
---|
1271 | (when (and (endp (cdr (poly-with-sugar-poly (elt G (first p)))))
|
---|
1272 | (endp (cdr (poly-with-sugar-poly (elt G (second p))))))
|
---|
1273 | ;;Record the pair as treated
|
---|
1274 | (setf (gethash p M) t)
|
---|
1275 | t))
|
---|
1276 | C))
|
---|
1277 | (labels ((pair-sugar (p)
|
---|
1278 | (spoly-sugar (elt G (first p)) (elt G (second p)) ring))
|
---|
1279 | (pair-mock-spoly (p)
|
---|
1280 | (mock-spoly (poly-with-sugar-poly (elt G (first p)))
|
---|
1281 | (poly-with-sugar-poly (elt G (second p))) pred))
|
---|
1282 | (pair-order (p q)
|
---|
1283 | (let ((sugar-p (third p))
|
---|
1284 | (sugar-q (third q)))
|
---|
1285 | (cond
|
---|
1286 | ((< sugar-p sugar-q) t)
|
---|
1287 | ((> sugar-p sugar-q) nil)
|
---|
1288 | (t
|
---|
1289 | (funcall pred (fourth q) (fourth p)))))))
|
---|
1290 | ;;Add sugar and mock S-polynomial as the third and fourth element
|
---|
1291 | ;;of new pair in C
|
---|
1292 | (dolist (p C)
|
---|
1293 | (setf (cdr (last p))
|
---|
1294 | (list (pair-sugar p) (pair-mock-spoly p))))
|
---|
1295 | (merge 'list B (sort C #'pair-order) #'pair-order)))
|
---|
1296 |
|
---|
1297 | (defun buchberger-with-sugar-sort-pairs (C G M pred ring)
|
---|
1298 | "Sorts critical pairs C according to sugar strategy."
|
---|
1299 | (buchberger-with-sugar-merge-pairs nil C G M pred ring))
|
---|
1300 |
|
---|
1301 | (defun Criterion-1-with-sugar (pair G &aux (i (first pair)) (j (second pair)))
|
---|
1302 | "An implementation of Buchberger's first criterion for polynomials
|
---|
1303 | with sugar. See the documentation of BUCHBERGER-CRITERION-1."
|
---|
1304 | (monom-rel-prime (poly-with-sugar-lm (elt G i))
|
---|
1305 | (poly-with-sugar-lm (elt G j))))
|
---|
1306 |
|
---|
1307 | (defun Criterion-2-with-sugar (pair G M &aux (i (first pair)) (j (second pair)))
|
---|
1308 | "An implementation of Buchberger's first criterion for polynomials
|
---|
1309 | with sugar. See the documentation of BUCHBERGER-CRITERION-2."
|
---|
1310 | (labels ((make-pair (u v) (if (< u v) (list u v) (list v u))))
|
---|
1311 | (dotimes (k (length G) nil)
|
---|
1312 | (when (and (/= k i)
|
---|
1313 | (/= k j)
|
---|
1314 | (monom-divides-p
|
---|
1315 | (poly-with-sugar-lm (elt G k))
|
---|
1316 | (monom-lcm (poly-with-sugar-lm (elt G i))
|
---|
1317 | (poly-with-sugar-lm (elt G j))))
|
---|
1318 | (gethash (make-pair i k) M)
|
---|
1319 | (gethash (make-pair k j) M))
|
---|
1320 | #+debug(when *grobner-debug* (format t ":B2"))
|
---|
1321 | (return-from Criterion-2-with-sugar t)))))
|
---|
1322 |
|
---|
1323 |
|
---|
1324 | ;;----------------------------------------------------------------
|
---|
1325 | ;; An implementation of the algorithm of Gebauer and Moeller,
|
---|
1326 | ;; as described in the book of Becker-Weispfenning, p. 232
|
---|
1327 | ;; with sugar strategy
|
---|
1328 | ;;----------------------------------------------------------------
|
---|
1329 | (defun gebauer-moeller-with-sugar (F pred start top-reduction-only ring
|
---|
1330 | &aux B G F1)
|
---|
1331 | "Compute Grobner basis by using the algorithm of Gebauer and Moeller.
|
---|
1332 | This algorithm is described as BUCHBERGERNEW2 in the book by
|
---|
1333 | Becker-Weispfenning entitled ``Grobner Bases''"
|
---|
1334 | ;;cut startup costs if (length F)<=1
|
---|
1335 | (cond
|
---|
1336 | ((endp F) (return-from gebauer-moeller-with-sugar nil))
|
---|
1337 | ((endp (cdr F))
|
---|
1338 | (return-from gebauer-moeller-with-sugar (list (grobner-primitive-part (car F) ring)))))
|
---|
1339 | #+debug (debug-cgb "~&GROBNER BASIS - GEBAUER MOELLER ALGORITHM WITH SUGAR STRATEGY")
|
---|
1340 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
|
---|
1341 | #+grobner-check (when (plusp start)
|
---|
1342 | (grobner-test (subseq F 0 start) (subseq F 0 start)
|
---|
1343 | pred ring))
|
---|
1344 | (setf B nil
|
---|
1345 | G (subseq F 0 start)
|
---|
1346 | F1 (subseq F start))
|
---|
1347 | ;;Initialize sugar
|
---|
1348 | (setf G (mapcar #'(lambda (x) (poly-add-sugar x ring)) G)
|
---|
1349 | F1 (mapcar #'(lambda (x) (poly-add-sugar x ring)) F1))
|
---|
1350 | (do () ((endp F1))
|
---|
1351 | (multiple-value-setq (G B)
|
---|
1352 | (update-with-sugar G B (grobner-primitive-part-with-sugar (pop F1) ring) pred ring)))
|
---|
1353 | (do ()
|
---|
1354 | ((endp B))
|
---|
1355 | (let* ((pair (pop B))
|
---|
1356 | (g1 (first pair))
|
---|
1357 | (g2 (second pair))
|
---|
1358 | (h (normal-form-with-sugar (spoly-with-sugar g1 g2 pred ring)
|
---|
1359 | G pred top-reduction-only ring)))
|
---|
1360 | (cond
|
---|
1361 | ((poly-with-sugar-zerop h))
|
---|
1362 | (t
|
---|
1363 | (setf h (grobner-primitive-part-with-sugar h ring))
|
---|
1364 | (multiple-value-setq (G B)
|
---|
1365 | (update-with-sugar G B h pred ring))
|
---|
1366 | #+debug (debug-cgb "~&[[~d]] Polynomials: ~d; Pairs left: ~d~%"
|
---|
1367 | (poly-with-sugar-sugar h)
|
---|
1368 | (length G) (length B))
|
---|
1369 | ))))
|
---|
1370 | (setf G (mapcar #'poly-with-sugar-poly G))
|
---|
1371 | ;;#+grobner-check(grobner-test G F pred ring)
|
---|
1372 | #+debug(debug-cgb "~&GROBNER END")
|
---|
1373 | G)
|
---|
1374 |
|
---|
1375 | (defun update-with-sugar (G B h pred ring &aux C D E B-new G-new pair)
|
---|
1376 | "An implementation of the auxillary UPDATE algorithm used by the
|
---|
1377 | Gebauer-Moeller algorithm. G is a list of polynomials, B is a list of
|
---|
1378 | critical pairs and H is a new polynomial which possibly will be added
|
---|
1379 | to G. The naming conventions used are very close to the one used in
|
---|
1380 | the book of Becker-Weispfenning. Operates on polynomials with sugar."
|
---|
1381 | (setf C G D nil)
|
---|
1382 | (do (g1) ((endp C))
|
---|
1383 | (setf g1 (pop C))
|
---|
1384 | (when (or (monom-rel-prime (poly-with-sugar-lm h) (poly-with-sugar-lm g1))
|
---|
1385 | (and
|
---|
1386 | (not (some
|
---|
1387 | #'(lambda (g2) (monom-divides-p (monom-lcm (poly-with-sugar-lm h)
|
---|
1388 | (poly-with-sugar-lm g2))
|
---|
1389 | (monom-lcm (poly-with-sugar-lm h)
|
---|
1390 | (poly-with-sugar-lm g1))))
|
---|
1391 | C))
|
---|
1392 | (not (some
|
---|
1393 | #'(lambda (g2) (monom-divides-p (monom-lcm (poly-with-sugar-lm h)
|
---|
1394 | (poly-with-sugar-lm g2))
|
---|
1395 | (monom-lcm (poly-with-sugar-lm h)
|
---|
1396 | (poly-with-sugar-lm g1))))
|
---|
1397 | D))))
|
---|
1398 | (push g1 D)))
|
---|
1399 | (setf E nil)
|
---|
1400 | (do (g1) ((endp D))
|
---|
1401 | (setf g1 (pop D))
|
---|
1402 | (unless (monom-rel-prime (poly-with-sugar-lm h) (poly-with-sugar-lm g1))
|
---|
1403 | (push g1 E)))
|
---|
1404 | (setf B-new nil)
|
---|
1405 | (do (g1 g2) ((endp B))
|
---|
1406 | (setf pair (pop B)
|
---|
1407 | g1 (first pair)
|
---|
1408 | g2 (second pair))
|
---|
1409 | (when (or (not (monom-divides-p (poly-with-sugar-lm h) (monom-lcm (poly-with-sugar-lm g1)
|
---|
1410 | (poly-with-sugar-lm g2))))
|
---|
1411 | (equal (monom-lcm (poly-with-sugar-lm g1) (poly-with-sugar-lm h))
|
---|
1412 | (monom-lcm (poly-with-sugar-lm g1) (poly-with-sugar-lm g2)))
|
---|
1413 | (equal (monom-lcm (poly-with-sugar-lm h) (poly-with-sugar-lm g2))
|
---|
1414 | (monom-lcm (poly-with-sugar-lm g1) (poly-with-sugar-lm g2))))
|
---|
1415 | (setf B-new (gebauer-moeller-with-sugar-merge-pairs B-new (list (list g1 g2)) pred ring))))
|
---|
1416 | (dolist (g3 E)
|
---|
1417 | (setf B-new (gebauer-moeller-with-sugar-merge-pairs B-new (list (list h g3)) pred ring)))
|
---|
1418 | (setf G-new nil)
|
---|
1419 | (do (g1) ((endp G))
|
---|
1420 | (setf g1 (pop G))
|
---|
1421 | (unless (monom-divides-p (poly-with-sugar-lm h) (poly-with-sugar-lm g1))
|
---|
1422 | (push g1 G-new)))
|
---|
1423 | (push h G-new)
|
---|
1424 | (values G-new B-new))
|
---|
1425 |
|
---|
1426 | (defun gebauer-moeller-with-sugar-merge-pairs (B C pred ring)
|
---|
1427 | "Merges lists of critical pairs. It orders pairs according to
|
---|
1428 | increasing sugar, with ties broken by smaller lcm of head monomials In
|
---|
1429 | this function B must already be sorted. Operates on polynomials with sugar."
|
---|
1430 | (setf C (delete-if
|
---|
1431 | #'(lambda (p) (and (endp (cdr (first p)))
|
---|
1432 | (endp (cdr (second p)))))
|
---|
1433 | C))
|
---|
1434 | (labels ((pair-sugar (p)
|
---|
1435 | (spoly-sugar (first p) (second p) ring))
|
---|
1436 | (pair-mock-spoly (p)
|
---|
1437 | (mock-spoly (poly-with-sugar-poly (first p))
|
---|
1438 | (poly-with-sugar-poly (second p)) pred))
|
---|
1439 | (pair-order (p q)
|
---|
1440 | (let ((sugar-p (pair-sugar p))
|
---|
1441 | (sugar-q (pair-sugar q)))
|
---|
1442 | (cond
|
---|
1443 | ((< sugar-p sugar-q) t)
|
---|
1444 | ((> sugar-p sugar-q) nil)
|
---|
1445 | (t
|
---|
1446 | (funcall pred (pair-mock-spoly q) (pair-mock-spoly p)))))))
|
---|
1447 | (merge 'list B (sort C #'pair-order) #'pair-order)))
|
---|
1448 |
|
---|
1449 | (defun grobner-primitive-part-with-sugar (h ring)
|
---|
1450 | (cons (grobner-primitive-part (car h) ring) (cdr h)))
|
---|
1451 |
|
---|
1452 | ;;----------------------------------------------------------------
|
---|
1453 | ;;
|
---|
1454 | ;; An implementation of the algorithms using product caching
|
---|
1455 | ;;
|
---|
1456 | ;;----------------------------------------------------------------
|
---|
1457 | (defun cached-normal-form (f fl pred top-reduction-only ring cache)
|
---|
1458 | "This version of NORMAL-FORM uses a cache which stores products of monomials
|
---|
1459 | and polynomials found in FL. The cache is a hash table storing pairs (G . G-HASH-TABLE)
|
---|
1460 | where G is a polynomial in FL serving as a key. The hash table G-HASH-TABLE is a list
|
---|
1461 | of pairs (MONOMIAL . PRODUCT-POLY) where PRODUCT-POLY is (POLY* MONOMIAL G)."
|
---|
1462 |
|
---|
1463 | (declare (optimize (speed 3) (safety 0)))
|
---|
1464 | ;; Loop invariant: c*f=sum ai*fi+r+p
|
---|
1465 | (do (r (c (ring-unit ring))
|
---|
1466 | (division-count 0)
|
---|
1467 | (p f))
|
---|
1468 | ((or (endp p)
|
---|
1469 | (endp fl)
|
---|
1470 | (and top-reduction-only r))
|
---|
1471 | #+debug(progn
|
---|
1472 | (debug-cgb "~&~3T~d reduction~:p" division-count)
|
---|
1473 | (when (endp r)
|
---|
1474 | (debug-cgb " ---> 0")))
|
---|
1475 | (values (append (nreverse r) p)
|
---|
1476 | division-count c))
|
---|
1477 | (declare (fixnum division-count) (integer c))
|
---|
1478 | (let ((i (position (lm p) fl :test #'monom-divisible-by-p :key #'lm)))
|
---|
1479 | (cond
|
---|
1480 | (i ;division possible
|
---|
1481 | (incf division-count)
|
---|
1482 | (let* ((g (nth i fl))
|
---|
1483 | (hash-tab (nth i cache))
|
---|
1484 | (gcd (funcall (ring-gcd ring) (lc g) (lc p)))
|
---|
1485 | (c1 (funcall (ring-/ ring) (lc g) gcd))
|
---|
1486 | (c2 (funcall (ring-/ ring) (lc p) gcd))
|
---|
1487 | (m (monom/ (lm p) (lm g))))
|
---|
1488 | ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
|
---|
1489 | (setf r (scalar-times-poly c1 r ring)
|
---|
1490 | c (funcall (ring-* ring) c c1)
|
---|
1491 | p (cached-grobner-op c2 c1 m p g pred ring hash-tab))))
|
---|
1492 | (t ;no division possible
|
---|
1493 | (setf r (cons (lt p) r) ;move lt(p) to remainder
|
---|
1494 | p (rest p)))))))
|
---|
1495 |
|
---|
1496 | (defun cached-grobner-op (c1 c2 m A B pred ring cache)
|
---|
1497 | "Returns the same result as (GROBNER-OP C1 C2 M A B PRED RING) but
|
---|
1498 | it used variable CACHE which is described in CACHED-NORMAL-FORM for
|
---|
1499 | efficient calculation of products of monomials with polynomials."
|
---|
1500 | (poly- (scalar-times-poly c2 A)
|
---|
1501 | (scalar-times-poly c1 (cached-monom-times-poly m B cache) ring)
|
---|
1502 | pred ring))
|
---|
1503 |
|
---|
1504 | (defun cached-buchberger (F pred start top-reduction-only ring
|
---|
1505 | &aux (s (1- (length F))) B M CACHE)
|
---|
1506 | "It returns the same result as BUCHBERGER but it uses caching
|
---|
1507 | of products of monomials with polynomials."
|
---|
1508 | (declare (fixnum s))
|
---|
1509 | (unless (plusp s) (return-from cached-buchberger F)) ;cut startup costs
|
---|
1510 | #+debug (debug-cgb "~&GROBNER BASIS - CACHED BUCHBERGER ALGORITHM")
|
---|
1511 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
|
---|
1512 | #+grobner-check (when (plusp start)
|
---|
1513 | (grobner-test (subseq F 0 start) (subseq F 0 start) pred ring))
|
---|
1514 | (setf B (nconc (makelist (list i j) (i 0 (1- start)) (j start s))
|
---|
1515 | (makelist (list i j) (i start (1- s)) (j (1+ i) s)))
|
---|
1516 | M (make-hash-table :test #'equal)
|
---|
1517 | B (buchberger-sort-pairs B F pred ring))
|
---|
1518 | ;;Initialize cache
|
---|
1519 | (dotimes (k (1+ s)) (push (make-hash-table :test #'equal) CACHE))
|
---|
1520 | ;;Initialize treated pairs
|
---|
1521 | (dotimes (i (1- start))
|
---|
1522 | (do ((j (1+ i) (1+ j))) ((>= j start))
|
---|
1523 | (setf (gethash (list i j) M) t)))
|
---|
1524 | (do ((G F))
|
---|
1525 | ((endp B)
|
---|
1526 | #+grobner-check(grobner-test G F pred ring)
|
---|
1527 | #+debug(debug-cgb "~&GROBNER END")
|
---|
1528 | G)
|
---|
1529 | (let ((pair (pop B)))
|
---|
1530 | (unless (or (Criterion-1 pair G)
|
---|
1531 | (Criterion-2 pair G M))
|
---|
1532 | (let ((SP (cached-normal-form (cached-spoly (elt G (first pair))
|
---|
1533 | (elt G (second pair))
|
---|
1534 | pred
|
---|
1535 | ring
|
---|
1536 | (elt CACHE (first pair))
|
---|
1537 | (elt CACHE (second pair)))
|
---|
1538 | G pred top-reduction-only ring CACHE)))
|
---|
1539 | (unless (poly-zerop SP) ;S-polynomial ---> 0;
|
---|
1540 | (setf s (1+ s)
|
---|
1541 | SP (grobner-primitive-part SP ring)
|
---|
1542 | (cdr (last G)) (list SP)
|
---|
1543 | (cdr (last Cache)) (list (make-hash-table :test #'equal))
|
---|
1544 | B (funcall *buchberger-merge-pairs* B (makelist (list i s)
|
---|
1545 | (i 0 (1- s)))
|
---|
1546 | G pred ring))
|
---|
1547 | #+debug (debug-cgb "~&Polynomials: ~d; Pairs left: ~d; Treated pairs: ~d"
|
---|
1548 | (length G) (length B)
|
---|
1549 | (hash-table-count M)))))
|
---|
1550 | (setf (gethash pair M) t))))
|
---|
1551 |
|
---|
1552 | (defun cached-spoly (f g pred ring cache-f cache-g)
|
---|
1553 | "Uses CACHE to fetch products of monomials and polynomials."
|
---|
1554 | (declare (optimize (speed 3) (safety 0)))
|
---|
1555 | (let* ((lcm (monom-lcm (lm f) (lm g)))
|
---|
1556 | (m1 (monom/ lcm (lm f)))
|
---|
1557 | (m2 (monom/ lcm (lm g))))
|
---|
1558 | (let* ((c (funcall (ring-lcm ring) (lc f) (lc g)))
|
---|
1559 | (cf (funcall (ring-/ ring) c (lc f)))
|
---|
1560 | (cg (funcall (ring-/ ring) c (lc g))))
|
---|
1561 | (poly-
|
---|
1562 | (scalar-times-poly cf (cached-monom-times-poly m1 f cache-f) ring)
|
---|
1563 | (scalar-times-poly cg (cached-monom-times-poly m2 g cache-g) ring)
|
---|
1564 | pred
|
---|
1565 | ring))))
|
---|
1566 |
|
---|
1567 |
|
---|
1568 | (defun cached-monom-times-poly (m f cache)
|
---|
1569 | (let ((prod (gethash m cache)))
|
---|
1570 | (cond
|
---|
1571 | (prod #+debug(debug-cgb "c") prod)
|
---|
1572 | (t
|
---|
1573 | #+debug(debug-cgb ".")
|
---|
1574 | (setf (gethash m cache) (monom-times-poly m f))))))
|
---|
1575 |
|
---|
1576 | ;;------------------------------------------------------------------------------------------------
|
---|
1577 | ;;
|
---|
1578 | ;; An implementation of the sugar strategy with product caching
|
---|
1579 | ;;
|
---|
1580 | ;;------------------------------------------------------------------------------------------------
|
---|
1581 | (defun cached-normal-form-with-sugar (f fl pred top-reduction-only ring cache)
|
---|
1582 | "Normal form of the polynomial with sugar F with respect to
|
---|
1583 | a list of polynomials with sugar FL. Assumes that FL is not empty.
|
---|
1584 | Parameters PRED, TOP-REDUCTION-ONLY and RING have the same meaning as in NORMAL-FORM.
|
---|
1585 | The parameter SUGAR-LIMIT should be set to a positive integer. If the sugar limit of
|
---|
1586 | the partial remainder exceeds SUGAR-LIMIT then the calculation is stopped and the partial
|
---|
1587 | remainder is returned, although it is not fully reduced with respect to FL."
|
---|
1588 | (declare (optimize (speed 3) (safety 0)))
|
---|
1589 | ;; Loop invariant: c*f=sum ai*fi+r+p
|
---|
1590 | (do ((r (cons nil 0))
|
---|
1591 | (c (ring-unit ring))
|
---|
1592 | (division-count 0)
|
---|
1593 | (p f))
|
---|
1594 | ((or (poly-with-sugar-zerop p)
|
---|
1595 | (and top-reduction-only (not (poly-with-sugar-zerop r))))
|
---|
1596 | #+debug
|
---|
1597 | (progn
|
---|
1598 | (debug-cgb "~&~3T~d reduction~:p" division-count)
|
---|
1599 | (cond
|
---|
1600 | ((poly-with-sugar-zerop r)
|
---|
1601 | (debug-cgb " ---> 0"))))
|
---|
1602 | (values (poly-with-sugar-append (poly-with-sugar-nreverse r) p)
|
---|
1603 | division-count c))
|
---|
1604 | (declare (fixnum division-count) (integer c))
|
---|
1605 | (let ((i (position (poly-with-sugar-lm p) fl
|
---|
1606 | :test #'monom-divisible-by-p
|
---|
1607 | :key #'poly-with-sugar-lm)))
|
---|
1608 | (cond
|
---|
1609 | (i ;division possible
|
---|
1610 | (incf division-count)
|
---|
1611 | (let* ((g (nth i fl))
|
---|
1612 | (hash-tab (nth i cache))
|
---|
1613 | (gcd (funcall (ring-gcd ring) (poly-with-sugar-lc g) (poly-with-sugar-lc p)))
|
---|
1614 | (c1 (funcall (ring-/ ring) (poly-with-sugar-lc g) gcd))
|
---|
1615 | (c2 (funcall (ring-/ ring) (poly-with-sugar-lc p) gcd))
|
---|
1616 | (m (monom/ (poly-with-sugar-lm p) (poly-with-sugar-lm g))))
|
---|
1617 | ;; Multiply the equation c*f=sum ai*fi+r+p by c1.
|
---|
1618 | (setf r (scalar-times-poly-with-sugar c1 r ring)
|
---|
1619 | c (funcall (ring-* ring) c c1)
|
---|
1620 | p (scalar-times-poly-with-sugar c1 p ring)
|
---|
1621 | p (cached-poly-with-sugar-op p c2 m g pred ring hash-tab))))
|
---|
1622 | (t ;no division possible
|
---|
1623 | ;;move lt(p) to remainder
|
---|
1624 | (setf r (cons (cons (poly-with-sugar-lt p) (poly-with-sugar-poly r))
|
---|
1625 | (poly-with-sugar-sugar r))
|
---|
1626 | p (poly-with-sugar-tail p)))))))
|
---|
1627 |
|
---|
1628 | ;; Parameter START is used when Grobner basis is calculated incrementally
|
---|
1629 | ;; and it signals that the elements of F from 0 to START-1 form a Grobner
|
---|
1630 | ;; basis already, so their pairs are not formed.
|
---|
1631 | (defun cached-buchberger-with-sugar (F-no-sugar pred start top-reduction-only ring
|
---|
1632 | &aux (s (1- (length F-no-sugar))) B M F Cache)
|
---|
1633 | "Returns a Grobner basis of an ideal generated by a list of ordinary
|
---|
1634 | polynomials F-NO-SUGAR. Adds initial sugar to the polynomials and
|
---|
1635 | performs the Buchberger algorithm with ``sugar strategy''. It returns
|
---|
1636 | an ordinary list of polynomials with no sugar. One of the most
|
---|
1637 | effective algorithms for Grobner basis calculation."
|
---|
1638 | (declare (fixnum s))
|
---|
1639 | (unless (plusp s) (return-from cached-buchberger-with-sugar F-no-sugar)) ;cut startup costs
|
---|
1640 | #+debug (debug-cgb "~&GROBNER BASIS - BUCHBERGER WITH SUGER STRATEGY ALGORITHM")
|
---|
1641 | #+debug (when (plusp start) (debug-cgb "~&INCREMENTAL:~d done" start))
|
---|
1642 | #+grobner-check (when (plusp start)
|
---|
1643 | (grobner-test (subseq F-no-sugar 0 start)
|
---|
1644 | (subseq F-no-sugar 0 start) pred ring))
|
---|
1645 | (setf F (mapcar #'(lambda (x) (poly-add-sugar x ring)) F-no-sugar)
|
---|
1646 | B (nconc (makelist (list i j) (i 0 (1- start)) (j start s))
|
---|
1647 | (makelist (list i j) (i start (1- s)) (j (1+ i) s)))
|
---|
1648 | M (make-hash-table :test #'equal))
|
---|
1649 | ;;Initialize cache
|
---|
1650 | (dotimes (k (1+ s)) (push (make-hash-table :test #'equal) Cache))
|
---|
1651 | ;;Initialize treated pairs
|
---|
1652 | (dotimes (i (1- start))
|
---|
1653 | (do ((j (1+ i) (1+ j))) ((>= j start))
|
---|
1654 | (setf (gethash (list i j) M) t)))
|
---|
1655 | (setf B (buchberger-with-sugar-sort-pairs B F M pred ring))
|
---|
1656 | (do ((G F))
|
---|
1657 | ((endp B)
|
---|
1658 | (setf G (mapcar #'poly-with-sugar-poly G))
|
---|
1659 | ;;#+grobner-check(grobner-test G F-no-sugar pred ring)
|
---|
1660 | #+debug(debug-cgb "~&GROBNER END")
|
---|
1661 | G)
|
---|
1662 | (let ((pair (pop B)))
|
---|
1663 | (cond
|
---|
1664 | ((Criterion-1-with-sugar pair G))
|
---|
1665 | ((Criterion-2-with-sugar pair G M))
|
---|
1666 | (t
|
---|
1667 | (let ((SP (cached-normal-form-with-sugar
|
---|
1668 | (cons (cached-spoly (poly-with-sugar-poly (elt G (first pair)))
|
---|
1669 | (poly-with-sugar-poly (elt G (second pair)))
|
---|
1670 | pred ring
|
---|
1671 | (elt Cache (first pair))
|
---|
1672 | (elt Cache (second pair)))
|
---|
1673 | (third pair))
|
---|
1674 | G pred top-reduction-only ring Cache)))
|
---|
1675 | (cond
|
---|
1676 | ((poly-with-sugar-zerop SP))
|
---|
1677 | (t
|
---|
1678 | (setf s (1+ s)
|
---|
1679 | (poly-with-sugar-poly SP) (grobner-primitive-part
|
---|
1680 | (poly-with-sugar-poly SP) ring)
|
---|
1681 | (cdr (last G)) (list SP)
|
---|
1682 | (cdr (last Cache)) (list (make-hash-table :test #'equal))
|
---|
1683 | B (buchberger-with-sugar-merge-pairs B (makelist (list i s)
|
---|
1684 | (i 0 (1- s)))
|
---|
1685 | G M pred ring))
|
---|
1686 | #+debug (debug-cgb "~&[~d] Polynomials: ~d; Pairs left: ~d; Treated pairs: ~d"
|
---|
1687 | (third pair)
|
---|
1688 | (length G) (length B)
|
---|
1689 | (hash-table-count M)))))))
|
---|
1690 | (setf (cddr pair) nil
|
---|
1691 | (gethash pair M) t))))
|
---|
1692 |
|
---|
1693 | ;; Implement f-term*g
|
---|
1694 | (defun cached-poly-with-sugar-op (f c m g pred ring Cache)
|
---|
1695 | (poly-with-sugar- f (scalar-times-poly-with-sugar c (cached-monom-times-poly-with-sugar m g Cache) ring) pred ring))
|
---|
1696 |
|
---|
1697 | (defun cached-monom-times-poly-with-sugar (m f Cache)
|
---|
1698 | (cons
|
---|
1699 | (cached-monom-times-poly m (poly-with-sugar-poly f) Cache)
|
---|
1700 | (+ (poly-with-sugar-sugar f) (monom-sugar m))))
|
---|