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source: branches/f4grobner/ideal.lisp@ 506

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[73]1;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*-
2;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3;;;
4;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
5;;;
6;;; This program is free software; you can redistribute it and/or modify
7;;; it under the terms of the GNU General Public License as published by
8;;; the Free Software Foundation; either version 2 of the License, or
9;;; (at your option) any later version.
10;;;
11;;; This program is distributed in the hope that it will be useful,
12;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
13;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14;;; GNU General Public License for more details.
15;;;
16;;; You should have received a copy of the GNU General Public License
17;;; along with this program; if not, write to the Free Software
18;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
19;;;
20;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
21
[67]22;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
23;;
24;; Operations in ideal theory
25;;
26;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
27
[502]28(defpackage "IDEAL"
[505]29 (:use :cl :monomial :order :term :polynomial)
[502]30 (:export))
31
32(in-package :ideal)
33
[67]34;; Does the term depend on variable K?
35(defun term-depends-p (term k)
36 "Return T if the term TERM depends on variable number K."
37 (monom-depends-p (term-monom term) k))
38
39;; Does the polynomial P depend on variable K?
40(defun poly-depends-p (p k)
41 "Return T if the term polynomial P depends on variable number K."
42 (some #'(lambda (term) (term-depends-p term k)) (poly-termlist p)))
43
44(defun ring-intersection (plist k)
45 "This function assumes that polynomial list PLIST is a Grobner basis
46and it calculates the intersection with the ring R[x[k+1],...,x[n]], i.e.
47it discards polynomials which depend on variables x[0], x[1], ..., x[k]."
48 (dotimes (i k plist)
49 (setf plist
50 (remove-if #'(lambda (p)
51 (poly-depends-p p i))
52 plist))))
53
54(defun elimination-ideal (ring flist k
55 &optional (top-reduction-only $poly_top_reduction_only) (start 0)
56 &aux (*monomial-order*
57 (or *elimination-order*
58 (elimination-order k))))
59 (ring-intersection (reduced-grobner ring flist start top-reduction-only) k))
60
61(defun colon-ideal (ring f g &optional (top-reduction-only $poly_top_reduction_only))
62 "Returns the reduced Grobner basis of the colon ideal Id(F):Id(G),
63where F and G are two lists of polynomials. The colon ideal I:J is
64defined as the set of polynomials H such that for all polynomials W in
65J the polynomial W*H belongs to I."
66 (cond
67 ((endp g)
68 ;;Id(G) consists of 0 only so W*0=0 belongs to Id(F)
69 (if (every #'poly-zerop f)
70 (error "First ideal must be non-zero.")
[156]71 (list (make-poly-from-termlist
[67]72 (list (make-term
73 (make-monom (monom-dimension (poly-lm (find-if-not #'poly-zerop f)))
74 :initial-element 0)
75 (funcall (ring-unit ring))))))))
76 ((endp (cdr g))
77 (colon-ideal-1 ring f (car g) top-reduction-only))
78 (t
79 (ideal-intersection ring
80 (colon-ideal-1 ring f (car g) top-reduction-only)
81 (colon-ideal ring f (rest g) top-reduction-only)
82 top-reduction-only))))
83
84(defun colon-ideal-1 (ring f g &optional (top-reduction-only $poly_top_reduction_only))
85 "Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}), where
86F is a list of polynomials and G is a polynomial."
87 (mapcar #'(lambda (x) (poly-exact-divide ring x g)) (ideal-intersection ring f (list g) top-reduction-only)))
88
89
90(defun ideal-intersection (ring f g &optional (top-reduction-only $poly_top_reduction_only)
91 &aux (*monomial-order* (or *elimination-order*
92 #'elimination-order-1)))
93 (mapcar #'poly-contract
94 (ring-intersection
95 (reduced-grobner
96 ring
97 (append (mapcar #'(lambda (p) (poly-extend p (make-monom 1 :initial-element 1))) f)
98 (mapcar #'(lambda (p)
99 (poly-append (poly-extend (poly-uminus ring p)
100 (make-monom 1 :initial-element 1))
101 (poly-extend p)))
102 g))
103 0
104 top-reduction-only)
105 1)))
106
107(defun poly-lcm (ring f g)
108 "Return LCM (least common multiple) of two polynomials F and G.
109The polynomials must be ordered according to monomial order PRED
110and their coefficients must be compatible with the RING structure
111defined in the COEFFICIENT-RING package."
112 (cond
113 ((poly-zerop f) f)
114 ((poly-zerop g) g)
115 ((and (endp (cdr (poly-termlist f))) (endp (cdr (poly-termlist g))))
116 (let ((m (monom-lcm (poly-lm f) (poly-lm g))))
117 (make-poly-from-termlist (list (make-term m (funcall (ring-lcm ring) (poly-lc f) (poly-lc g)))))))
118 (t
119 (multiple-value-bind (f f-cont)
120 (poly-primitive-part ring f)
121 (multiple-value-bind (g g-cont)
122 (poly-primitive-part ring g)
123 (scalar-times-poly
124 ring
125 (funcall (ring-lcm ring) f-cont g-cont)
126 (poly-primitive-part ring (car (ideal-intersection ring (list f) (list g) nil)))))))))
127
128;; Do two Grobner bases yield the same ideal?
129(defun grobner-equal (ring g1 g2)
130 "Returns T if two lists of polynomials G1 and G2, assumed to be Grobner bases,
131generate the same ideal, and NIL otherwise."
132 (and (grobner-subsetp ring g1 g2) (grobner-subsetp ring g2 g1)))
133
134(defun grobner-subsetp (ring g1 g2)
135 "Returns T if a list of polynomials G1 generates
136an ideal contained in the ideal generated by a polynomial list G2,
137both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise."
138 (every #'(lambda (p) (grobner-member ring p g2)) g1))
139
140(defun grobner-member (ring p g)
141 "Returns T if a polynomial P belongs to the ideal generated by the
142polynomial list G, which is assumed to be a Grobner basis. Returns NIL otherwise."
143 (poly-zerop (normal-form ring p g nil)))
144
145;; Calculate F : p^inf
146(defun ideal-saturation-1 (ring f p start &optional (top-reduction-only $poly_top_reduction_only)
147 &aux (*monomial-order* (or *elimination-order*
148 #'elimination-order-1)))
149 "Returns the reduced Grobner basis of the saturation of the ideal
150generated by a polynomial list F in the ideal generated by a single
151polynomial P. The saturation ideal is defined as the set of
152polynomials H such for some natural number n (* (EXPT P N) H) is in the ideal
153F. Geometrically, over an algebraically closed field, this is the set
154of polynomials in the ideal generated by F which do not identically
155vanish on the variety of P."
156 (mapcar
157 #'poly-contract
158 (ring-intersection
159 (reduced-grobner
160 ring
161 (saturation-extension-1 ring f p)
162 start top-reduction-only)
163 1)))
164
165
166
167;; Calculate F : p1^inf : p2^inf : ... : ps^inf
168(defun ideal-polysaturation-1 (ring f plist start &optional (top-reduction-only $poly_top_reduction_only))
169 "Returns the reduced Grobner basis of the ideal obtained by a
170sequence of successive saturations in the polynomials
171of the polynomial list PLIST of the ideal generated by the
172polynomial list F."
173 (cond
174 ((endp plist) (reduced-grobner ring f start top-reduction-only))
175 (t (let ((g (ideal-saturation-1 ring f (car plist) start top-reduction-only)))
176 (ideal-polysaturation-1 ring g (rest plist) (length g) top-reduction-only)))))
177
178(defun ideal-saturation (ring f g start &optional (top-reduction-only $poly_top_reduction_only)
179 &aux
180 (k (length g))
181 (*monomial-order* (or *elimination-order*
182 (elimination-order k))))
183 "Returns the reduced Grobner basis of the saturation of the ideal
184generated by a polynomial list F in the ideal generated a polynomial
185list G. The saturation ideal is defined as the set of polynomials H
186such for some natural number n and some P in the ideal generated by G
187the polynomial P**N * H is in the ideal spanned by F. Geometrically,
188over an algebraically closed field, this is the set of polynomials in
189the ideal generated by F which do not identically vanish on the
190variety of G."
191 (mapcar
192 #'(lambda (q) (poly-contract q k))
193 (ring-intersection
194 (reduced-grobner ring
195 (polysaturation-extension ring f g)
196 start
197 top-reduction-only)
198 k)))
199
200(defun ideal-polysaturation (ring f ideal-list start &optional (top-reduction-only $poly_top_reduction_only))
201 "Returns the reduced Grobner basis of the ideal obtained by a
202successive applications of IDEAL-SATURATION to F and lists of
203polynomials in the list IDEAL-LIST."
204 (cond
205 ((endp ideal-list) f)
206 (t (let ((h (ideal-saturation ring f (car ideal-list) start top-reduction-only)))
207 (ideal-polysaturation ring h (rest ideal-list) (length h) top-reduction-only)))))
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