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