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