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1;;; -*- Mode: Lisp -*-
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
22;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
23;;
24;; Operations in ideal theory
25;;
26;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
27
28(defpackage "IDEAL"
29 (:use :cl :ring :monom :order :term :polynomial :division :grobner-wrap :ring-and-order)
30 (:export "POLY-DEPENDS-P"
31 "RING-INTERSECTION"
32 "ELIMINATION-IDEAL"
33 "COLON-IDEAL"
34 "COLON-IDEAL-1"
35 "IDEAL-INTERSECTION"
36 "POLY-LCM"
37 "GROBNER-EQUAL"
38 "GROBNER-SUBSETP"
39 "GROBNER-MEMBER"
40 "IDEAL-SATURATION-1"
41 "IDEAL-SATURATION"
42 "IDEAL-POLYSATURATION-1"
43 "IDEAL-POLYSATURATION"
44 ))
45
46(in-package :ideal)
47
48;; Does the polynomial P depend on variable K?
49(defun poly-depends-p (p k)
50 "Return T if the term polynomial P depends on variable number K."
51 (some #'(lambda (term) (term-depends-p term k)) (poly-termlist p)))
52
53(defun ring-intersection (plist k)
54 "This function assumes that polynomial list
55PLIST=(P[0],P[1],...,P[J-1]) is a Grobner basis and it calculates the
56intersection of Id({P[0],P[1],...,P[J-1]}) with the ring
57R[X[K],...,X[N-1]], i.e. it discards polynomials which depend on
58variables X[0], X[1], ..., X[K-1]."
59 (dotimes (i k plist)
60 (setf plist
61 (remove-if #'(lambda (p)
62 (poly-depends-p p i))
63 plist))))
64
65(defun elimination-ideal (ring-and-order flist k
66 &optional
67 (top-reduction-only $poly_top_reduction_only)
68 (start 0))
69 "Given a list of polynomials FLIST, and an integer K, tt finds and
70returns the Groebner basis the elimination ideal of Id({FLIST})
71obtained by eliminating the first K variables. Optional argument
72TOP-REDUCTION-ONLY indicates whether to fully reduce or only
73top-reduce. Optional argument START, defaulting to 0, is used to
74indicate that the first START elements of F form a Groebner basis."
75 (ring-intersection (reduced-grobner ring-and-order flist start top-reduction-only) k))
76
77(defun colon-ideal (ring-and-order f g
78 &optional
79 (top-reduction-only $poly_top_reduction_only)
80 &aux
81 (ring (ro-ring ring-and-order)))
82 "Returns the reduced Grobner basis of the colon ideal Id(F):Id(G),
83where F and G are two lists of polynomials. The colon ideal I:J is
84defined as the set of polynomials H such that for all polynomials W in
85J the polynomial W*H belongs to I."
86 (declare (type ring-and-order ring-and-order))
87 (cond
88 ((endp g)
89 ;;Id(G) consists of 0 only so W*0=0 belongs to Id(F)
90 (if (every #'poly-zerop f)
91 (error "First ideal must be non-zero.")
92 (list (make-poly-from-termlist
93 (list (make-term
94 :monom (make-monom :dimension (monom-dimension (poly-lm (find-if-not #'poly-zerop f))))
95 :coeff (funcall (ring-unit ring))))))))
96 ((endp (cdr g))
97 (colon-ideal-1 ring-and-order f (car g) top-reduction-only))
98 (t
99 (ideal-intersection ring-and-order
100 (colon-ideal-1 ring-and-order f (car g) top-reduction-only)
101 (colon-ideal ring-and-order f (rest g) top-reduction-only)
102 top-reduction-only))))
103
104(defun colon-ideal-1 (ring-and-order f g
105 &optional
106 (top-reduction-only $poly_top_reduction_only))
107 "Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}), where
108F is a list of polynomials and G is a polynomial."
109 (declare (type ring-and-order ring-and-order))
110 (mapcar #'(lambda (x)
111 (poly-exact-divide ring-and-order x g))
112 (ideal-intersection ring-and-order f (list g) top-reduction-only)))
113
114(defun ideal-intersection (ring-and-order f g
115 &optional
116 (top-reduction-only $poly_top_reduction_only)
117 (ring (ro-ring ring-and-order)))
118 (declare (type ring-and-order ring-and-order))
119 (mapcar #'poly-contract
120 (ring-intersection
121 (reduced-grobner
122 ring-and-order
123 (append (mapcar #'(lambda (p) (poly-extend p (make-monom :dimension 1 :initial-exponent 1))) f)
124 (mapcar #'(lambda (p)
125 (poly-append (poly-extend (poly-uminus ring p)
126 (make-monom :dimension 1 :initial-exponent 1))
127 (poly-extend p)))
128 g))
129 0
130 top-reduction-only)
131 1)))
132
133(defun poly-lcm (ring-and-order f g &aux (ring (ro-ring ring-and-order)))
134 "Return LCM (least common multiple) of two polynomials F and G.
135The polynomials must be ordered according to monomial order PRED
136and their coefficients must be compatible with the RING structure
137defined in the COEFFICIENT-RING package."
138 (cond
139 ((poly-zerop f) f)
140 ((poly-zerop g) g)
141 ((and (endp (cdr (poly-termlist f))) (endp (cdr (poly-termlist g))))
142 (let ((m (monom-lcm (poly-lm f) (poly-lm g))))
143 (make-poly-from-termlist (list (make-term :monom m
144 :coeff (funcall (ring-lcm ring) (poly-lc f) (poly-lc g)))))))
145 (t
146 (multiple-value-bind (f f-cont)
147 (poly-primitive-part ring f)
148 (multiple-value-bind (g g-cont)
149 (poly-primitive-part ring g)
150 (scalar-times-poly
151 ring
152 (funcall (ring-lcm ring) f-cont g-cont)
153 (poly-primitive-part ring (car (ideal-intersection ring-and-order (list f) (list g) nil)))))))))
154
155;; Do two Grobner bases yield the same ideal?
156(defun grobner-equal (ring-and-order g1 g2)
157 "Returns T if two lists of polynomials G1 and G2, assumed to be Grobner bases,
158generate the same ideal, and NIL otherwise."
159 (declare (type ring-and-order ring-and-order))
160 (and (grobner-subsetp ring-and-order g1 g2) (grobner-subsetp ring-and-order g2 g1)))
161
162(defun grobner-subsetp (ring-and-order g1 g2)
163 "Returns T if a list of polynomials G1 generates
164an ideal contained in the ideal generated by a polynomial list G2,
165both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise."
166 (declare (type ring-and-order ring-and-order))
167 (every #'(lambda (p) (grobner-member ring-and-order p g2)) g1))
168
169(defun grobner-member (ring-and-order p g)
170 "Returns T if a polynomial P belongs to the ideal generated by the
171polynomial list G, which is assumed to be a Grobner basis. Returns NIL otherwise."
172 (declare (type ring-and-order ring-and-order))
173 (poly-zerop (normal-form ring-and-order p g nil)))
174
175;; Calculate F : p^inf
176(defun ideal-saturation-1 (ring-and-order f p
177 &optional
178 (start 0)
179 (top-reduction-only $poly_top_reduction_only)
180 &aux
181 (ring (ro-ring ring-and-order)))
182 "Returns the reduced Grobner basis of the saturation of the ideal
183generated by a polynomial list F in the ideal generated by a single
184polynomial P. The saturation ideal is defined as the set of
185polynomials H such for some natural number n (* (EXPT P N) H) is in
186the ideal spanned by F. Geometrically, over an algebraically closed
187field, this is the set of polynomials in the ideal generated by F
188which do not identically vanish on the variety of P."
189 (declare (type ring-and-order ring-and-order))
190 (mapcar
191 #'poly-contract
192 (ring-intersection
193 (reduced-grobner
194 ring-and-order
195 (saturation-extension-1 ring f p)
196 start top-reduction-only)
197 1)))
198
199
200;; Calculate F : p1^inf : p2^inf : ... : ps^inf
201(defun ideal-polysaturation-1 (ring-and-order f plist
202 &optional
203 (start 0)
204 (top-reduction-only $poly_top_reduction_only))
205 "Returns the reduced Grobner basis of the ideal obtained by a
206sequence of successive saturations in the polynomials
207of the polynomial list PLIST of the ideal generated by the
208polynomial list F."
209 (cond
210 ((endp plist)
211 (reduced-grobner ring-and-order f start top-reduction-only))
212 (t (let ((g (ideal-saturation-1 ring-and-order f (car plist) start top-reduction-only)))
213 (ideal-polysaturation-1 ring-and-order g (rest plist) (length g) top-reduction-only)))))
214
215(defun ideal-saturation (ring-and-order f g
216 &optional
217 (start 0)
218 (top-reduction-only $poly_top_reduction_only)
219 &aux
220 (k (length g))
221 (ring (ro-ring ring-and-order)))
222 "Returns the reduced Grobner basis of the saturation of the ideal
223generated by a polynomial list F in the ideal generated a polynomial
224list G. The saturation ideal is defined as the set of polynomials H
225such for some natural number n and some P in the ideal generated by G
226the polynomial P**N * H is in the ideal spanned by F. Geometrically,
227over an algebraically closed field, this is the set of polynomials in
228the ideal generated by F which do not identically vanish on the
229variety of G."
230 (declare (type ring-and-order ring-and-order))
231 (mapcar
232 #'(lambda (q) (poly-contract q k))
233 (ring-intersection
234 (reduced-grobner ring-and-order
235 (polysaturation-extension ring f g)
236 start
237 top-reduction-only)
238 k)))
239
240(defun ideal-polysaturation (ring-and-order f ideal-list
241 &optional
242 (start 0)
243 (top-reduction-only $poly_top_reduction_only))
244 "Returns the reduced Grobner basis of the ideal obtained by a
245successive applications of IDEAL-SATURATION to F and lists of
246polynomials in the list IDEAL-LIST."
247 (declare (type ring-and-order ring-and-order))
248 (cond
249 ((endp ideal-list) f)
250 (t (let ((h (ideal-saturation ring-and-order f (car ideal-list) start top-reduction-only)))
251 (ideal-polysaturation ring-and-order h (rest ideal-list) (length h) top-reduction-only)))))
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