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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 :monomial :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 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-and-order flist k &optional (top-reduction-only $poly_top_reduction_only) (start 0))
69 (ring-intersection (reduced-grobner ring-and-order flist start top-reduction-only) k))
70
71(defun colon-ideal (ring-and-order f g
72 &optional
73 (top-reduction-only $poly_top_reduction_only)
74 &aux
75 (ring (ro-ring ring-and-order)))
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 (declare (type ring-and-order ring-and-order))
81 (cond
82 ((endp g)
83 ;;Id(G) consists of 0 only so W*0=0 belongs to Id(F)
84 (if (every #'poly-zerop f)
85 (error "First ideal must be non-zero.")
86 (list (make-poly-from-termlist
87 (list (make-term
88 (make-monom :dimension (monom-dimension (poly-lm (find-if-not #'poly-zerop f))))
89 (funcall (ring-unit ring))))))))
90 ((endp (cdr g))
91 (colon-ideal-1 ring-and-order f (car g) top-reduction-only))
92 (t
93 (ideal-intersection ring-and-order
94 (colon-ideal-1 ring f (car g) top-reduction-only)
95 (colon-ideal ring-and-order f (rest g) top-reduction-only)
96 top-reduction-only))))
97
98(defun colon-ideal-1 (ring-and-order f g
99 &optional
100 (top-reduction-only $poly_top_reduction_only)
101 &aux
102 (ring (ro-ring ring-and-order)))
103 "Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}), where
104F is a list of polynomials and G is a polynomial."
105 (declare (type ring-and-order ring-and-order))
106 (mapcar #'(lambda (x)
107 (poly-exact-divide ring-and-order x g))
108 (ideal-intersection ring-and-order f (list g) top-reduction-only)))
109
110(defun ideal-intersection (ring-and-order f g
111 &optional
112 (top-reduction-only $poly_top_reduction_only)
113 &aux
114 (ring (ro-ring ring-and-order)))
115 (mapcar #'poly-contract
116 (ring-intersection
117 (reduced-grobner
118 ring-and-order
119 (append (mapcar #'(lambda (p) (poly-extend p (make-monom :dimension 1 :initial-exponent 1))) f)
120 (mapcar #'(lambda (p)
121 (poly-append (poly-extend (poly-uminus ring-and-order p)
122 (make-monom :dimension 1 :initial-exponent 1))
123 (poly-extend p)))
124 g))
125 0
126 top-reduction-only)
127 1)))
128
129(defun poly-lcm (ring-and-order f g &aux (ring (ro-ring ring-and-order)))
130 "Return LCM (least common multiple) of two polynomials F and G.
131The polynomials must be ordered according to monomial order PRED
132and their coefficients must be compatible with the RING structure
133defined in the COEFFICIENT-RING package."
134 (cond
135 ((poly-zerop f) f)
136 ((poly-zerop g) g)
137 ((and (endp (cdr (poly-termlist f))) (endp (cdr (poly-termlist g))))
138 (let ((m (monom-lcm (poly-lm f) (poly-lm g))))
139 (make-poly-from-termlist (list (make-term m (funcall (ring-lcm ring) (poly-lc f) (poly-lc g)))))))
140 (t
141 (multiple-value-bind (f f-cont)
142 (poly-primitive-part ring f)
143 (multiple-value-bind (g g-cont)
144 (poly-primitive-part ring g)
145 (scalar-times-poly
146 ring
147 (funcall (ring-lcm ring) f-cont g-cont)
148 (poly-primitive-part ring (car (ideal-intersection ring-and-order (list f) (list g) nil)))))))))
149
150;; Do two Grobner bases yield the same ideal?
151(defun grobner-equal (ring-and-order g1 g2)
152 "Returns T if two lists of polynomials G1 and G2, assumed to be Grobner bases,
153generate the same ideal, and NIL otherwise."
154 (declare (type ring-and-order ring-and-order))
155 (and (grobner-subsetp ring-and-order g1 g2) (grobner-subsetp ring-and-order g2 g1)))
156
157(defun grobner-subsetp (ring-and-order g1 g2)
158 "Returns T if a list of polynomials G1 generates
159an ideal contained in the ideal generated by a polynomial list G2,
160both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise."
161 (declare (type ring-and-order ring-and-order))
162 (every #'(lambda (p) (grobner-member ring-and-order p g2)) g1))
163
164(defun grobner-member (ring-and-order p g)
165 "Returns T if a polynomial P belongs to the ideal generated by the
166polynomial list G, which is assumed to be a Grobner basis. Returns NIL otherwise."
167 (declare (type ring-and-order ring-and-order))
168 (poly-zerop (normal-form ring-and-order p g nil)))
169
170;; Calculate F : p^inf
171(defun ideal-saturation-1 (ring-and-order f p start
172 &optional
173 (top-reduction-only $poly_top_reduction_only))
174 "Returns the reduced Grobner basis of the saturation of the ideal
175generated by a polynomial list F in the ideal generated by a single
176polynomial P. The saturation ideal is defined as the set of
177polynomials H such for some natural number n (* (EXPT P N) H) is in the ideal
178F. Geometrically, over an algebraically closed field, this is the set
179of polynomials in the ideal generated by F which do not identically
180vanish on the variety of P."
181 (declare (type ring-and-order ring-and-order))
182 (mapcar
183 #'poly-contract
184 (ring-intersection
185 (reduced-grobner
186 ring-and-order
187 (saturation-extension-1 ring-and-order f p)
188 start top-reduction-only)
189 1)))
190
191
192;; Calculate F : p1^inf : p2^inf : ... : ps^inf
193(defun ideal-polysaturation-1 (ring-and-order f plist start
194 &optional
195 (top-reduction-only $poly_top_reduction_only))
196 "Returns the reduced Grobner basis of the ideal obtained by a
197sequence of successive saturations in the polynomials
198of the polynomial list PLIST of the ideal generated by the
199polynomial list F."
200 (cond
201 ((endp plist)
202 (reduced-grobner ring-and-order f start top-reduction-only))
203 (t (let ((g (ideal-saturation-1 ring-and-order f (car plist) start top-reduction-only)))
204 (ideal-polysaturation-1 ring-and-order g (rest plist) (length g) top-reduction-only)))))
205
206(defun ideal-saturation (ring-and-order f g start &optional (top-reduction-only $poly_top_reduction_only)
207 &aux
208 (k (length g)))
209 "Returns the reduced Grobner basis of the saturation of the ideal
210generated by a polynomial list F in the ideal generated a polynomial
211list G. The saturation ideal is defined as the set of polynomials H
212such for some natural number n and some P in the ideal generated by G
213the polynomial P**N * H is in the ideal spanned by F. Geometrically,
214over an algebraically closed field, this is the set of polynomials in
215the ideal generated by F which do not identically vanish on the
216variety of G."
217 (mapcar
218 #'(lambda (q) (poly-contract q k))
219 (ring-intersection
220 (reduced-grobner ring-and-order
221 (polysaturation-extension ring-and-order f g)
222 start
223 top-reduction-only)
224 k)))
225
226(defun ideal-polysaturation (ring-and-order f ideal-list start
227 &optional
228 (top-reduction-only $poly_top_reduction_only))
229 "Returns the reduced Grobner basis of the ideal obtained by a
230successive applications of IDEAL-SATURATION to F and lists of
231polynomials in the list IDEAL-LIST."
232 (cond
233 ((endp ideal-list) f)
234 (t (let ((h (ideal-saturation ring-and-order f (car ideal-list) start top-reduction-only)))
235 (ideal-polysaturation ring-and-order h (rest ideal-list) (length h) top-reduction-only)))))
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