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source: branches/f4grobner/mx-grobner-0.lisp@ 1876

Last change on this file since 1876 was 1853, checked in by Marek Rychlik, 10 years ago

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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;; Load this file into Maxima to bootstrap the Grobner package.
25;; NOTE: This file does use symbols defined by Maxima, so it
26;; will not work when loaded in Common Lisp.
27;;
28;; DETAILS: This file implements an interface between the Grobner
29;; basis package NGROBNER, which is a pure Common Lisp package, and
30;; Maxima. NGROBNER for efficiency uses its own representation of
31;; polynomials. Thus, it is necessary to convert Maxima representation
32;; to the internal representation and back. The facilities to do so
33;; are implemented in this file.
34;;
35;; Also, since the NGROBNER package consists of many Lisp files, it is
36;; necessary to load the files. It is possible and preferrable to use
37;; ASDF for this purpose. The default is ASDF. It is also possible to
38;; simply used LOAD and COMPILE-FILE to accomplish this task.
39;;
40;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
41
42(in-package :maxima)
43
44(macsyma-module cgb-maxima)
45
46
47(eval-when
48 #+gcl (load eval)
49 #-gcl (:load-toplevel :execute)
50 (format t "~&Loading maxima-grobner ~a ~a~%"
51 "$Revision: 2.0 $" "$Date: 2015/06/02 0:34:17 $"))
52
53;;FUNCTS is loaded because it contains the definition of LCM
54($load "functs")
55#+sbcl(progn (require 'asdf) (load "ngrobner.asd")(asdf:load-system :ngrobner))
56
57(use-package :ngrobner)
58
59
60;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
61;;
62;; Maxima expression ring
63;;
64;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
65;;
66;; This is how we perform operations on coefficients
67;; using Maxima functions.
68;;
69;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
70
71(defparameter +maxima-ring+
72 (make-ring
73 ;;(defun coeff-zerop (expr) (meval1 `(($is) (($equal) ,expr 0))))
74 :parse #'(lambda (expr)
75 (when modulus (setf expr ($rat expr)))
76 expr)
77 :unit #'(lambda () (if modulus ($rat 1) 1))
78 :zerop #'(lambda (expr)
79 ;;When is exactly a maxima expression equal to 0?
80 (cond ((numberp expr)
81 (= expr 0))
82 ((atom expr) nil)
83 (t
84 (case (caar expr)
85 (mrat (eql ($ratdisrep expr) 0))
86 (otherwise (eql ($totaldisrep expr) 0))))))
87 :add #'(lambda (x y) (m+ x y))
88 :sub #'(lambda (x y) (m- x y))
89 :uminus #'(lambda (x) (m- x))
90 :mul #'(lambda (x y) (m* x y))
91 ;;(defun coeff-div (x y) (cadr ($divide x y)))
92 :div #'(lambda (x y) (m// x y))
93 :lcm #'(lambda (x y) (meval1 `((|$LCM|) ,x ,y)))
94 :ezgcd #'(lambda (x y) (apply #'values (cdr ($ezgcd ($totaldisrep x) ($totaldisrep y)))))
95 ;; :gcd #'(lambda (x y) (second ($ezgcd x y)))))
96 :gcd #'(lambda (x y) ($gcd x y))))
97
98;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
99;;
100;; Maxima expression parsing
101;;
102;;
103;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
104;;
105;; Functions and macros dealing with internal representation
106;; structure.
107;;
108;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
109
110(defun equal-test-p (expr1 expr2)
111 (alike1 expr1 expr2))
112
113(defun coerce-maxima-list (expr)
114 "Convert a Maxima list to Lisp list."
115 (cond
116 ((and (consp (car expr)) (eql (caar expr) 'mlist)) (cdr expr))
117 (t expr)))
118
119(defun free-of-vars (expr vars) (apply #'$freeof `(,@vars ,expr)))
120
121(defun parse-poly (expr vars &aux (vars (coerce-maxima-list vars)))
122 "Convert a maxima polynomial expression EXPR in variables VARS to internal form."
123 (labels ((parse (arg) (parse-poly arg vars))
124 (parse-list (args) (mapcar #'parse args)))
125 (cond
126 ((eql expr 0) (make-poly-zero))
127 ((member expr vars :test #'equal-test-p)
128 (let ((pos (position expr vars :test #'equal-test-p)))
129 (make-poly-variable +maxima-ring+ (length vars) pos)))
130 ((free-of-vars expr vars)
131 ;;This means that variable-free CRE and Poisson forms will be converted
132 ;;to coefficients intact
133 (coerce-coeff +maxima-ring+ expr vars))
134 (t
135 (case (caar expr)
136 (mplus (reduce #'(lambda (x y) (poly-add +maxima-ring+ x y)) (parse-list (cdr expr))))
137 (mminus (poly-uminus +maxima-ring+ (parse (cadr expr))))
138 (mtimes
139 (if (endp (cddr expr)) ;unary
140 (parse (cdr expr))
141 (reduce #'(lambda (p q) (poly-mul +maxima-ring+ p q)) (parse-list (cdr expr)))))
142 (mexpt
143 (cond
144 ((member (cadr expr) vars :test #'equal-test-p)
145 ;;Special handling of (expt var pow)
146 (let ((pos (position (cadr expr) vars :test #'equal-test-p)))
147 (make-poly-variable +maxima-ring+ (length vars) pos (caddr expr))))
148 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
149 ;; Negative power means division in coefficient ring
150 ;; Non-integer power means non-polynomial coefficient
151 (mtell "~%Warning: Expression ~%~M~%contains power which is not a positive integer. Parsing as coefficient.~%"
152 expr)
153 (coerce-coeff +maxima-ring+ expr vars))
154 (t (poly-expt +maxima-ring+ (parse (cadr expr)) (caddr expr)))))
155 (mrat (parse ($ratdisrep expr)))
156 (mpois (parse ($outofpois expr)))
157 (otherwise
158 (coerce-coeff +maxima-ring+ expr vars)))))))
159
160(defun parse-poly-list (expr vars)
161 "Parse a Maxima representation of a list of polynomials."
162 (case (caar expr)
163 (mlist (mapcar #'(lambda (p) (parse-poly p vars)) (cdr expr)))
164 (t (merror "Expression ~M is not a list of polynomials in variables ~M."
165 expr vars))))
166
167(defun parse-poly-list-list (poly-list-list vars)
168 "Parse a Maxima representation of a list of lists of polynomials."
169 (mapcar #'(lambda (g) (parse-poly-list g vars)) (coerce-maxima-list poly-list-list)))
170
171
172;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
173;;
174;; Conversion from internal form to Maxima general form
175;;
176;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
177
178(defun maxima-head ()
179 (if $poly_return_term_list
180 '(mlist)
181 '(mplus)))
182
183(defun coerce-to-maxima (poly-type object vars)
184 (case poly-type
185 (:polynomial
186 `(,(maxima-head) ,@(mapcar #'(lambda (term) (coerce-to-maxima :term term vars)) (poly-termlist object))))
187 (:poly-list
188 `((mlist) ,@(mapcar #'(lambda (p) (funcall *ratdisrep-fun* (coerce-to-maxima :polynomial p vars))) object)))
189 (:term
190 `((mtimes) ,(funcall *ratdisrep-fun* (term-coeff object))
191 ,@(mapcar #'(lambda (var power) `((mexpt) ,var ,power))
192 vars (coerce (term-monom object) 'list))))
193 ;; Assumes that Lisp and Maxima logicals coincide
194 (:logical object)
195 (otherwise
196 object)))
197
198
199;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
200;;
201;; Unary and binary operation definition facility
202;;
203;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
204
205(defmacro define-unop (maxima-name fun-name
206 &optional (documentation nil documentation-supplied-p))
207 "Define a MAXIMA-level unary operator MAXIMA-NAME corresponding to unary function FUN-NAME."
208 `(defun ,maxima-name (p vars
209 &aux
210 (vars (coerce-maxima-list vars))
211 (p (parse-poly p vars)))
212 ,@(when documentation-supplied-p (list documentation))
213 (coerce-to-maxima :polynomial (,fun-name +maxima-ring+ p) vars)))
214
215(defmacro define-binop (maxima-name fun-name
216 &optional (documentation nil documentation-supplied-p))
217 "Define a MAXIMA-level binary operator MAXIMA-NAME corresponding to binary function FUN-NAME."
218 `(defmfun ,maxima-name (p q vars
219 &aux
220 (vars (coerce-maxima-list vars))
221 (p (parse-poly p vars))
222 (q (parse-poly q vars)))
223 ,@(when documentation-supplied-p (list documentation))
224 (coerce-to-maxima :polynomial (,fun-name +maxima-ring+ p q) vars)))
225
226
227;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
228;;
229;; Order utilities
230;;
231;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
232
233(defun find-order (order)
234 "This function returns the order function bases on its name."
235 (cond
236 ((null order) nil)
237 ((symbolp order)
238 (case order
239 ((lex :lex $lex) #'lex>)
240 ((grlex :grlex $grlex) #'grlex>)
241 ((grevlex :grevlex $grevlex) #'grevlex>)
242 ((invlex :invlex $invlex) #'invlex>)
243 ((elimination-order-1 :elimination-order-1 elimination_order_1) #'elimination-order-1)
244 (otherwise
245 (mtell "~%Warning: Order ~M not found. Using default.~%" order))))
246 (t
247 (mtell "~%Order specification ~M is not recognized. Using default.~%" order)
248 nil)))
249
250;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
251;;
252;; Ring utilities
253;;
254;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
255
256(defun find-ring (ring)
257 "This function returns the ring structure bases on input symbol."
258 (cond
259 ((null ring) nil)
260 ((symbolp ring)
261 (case ring
262 ((maxima-ring :maxima-ring $expression_ring) +maxima-ring+)
263 ((ring-of-integers :ring-of-integers $ring_of_integers) +ring-of-integers+)
264 (otherwise
265 (mtell "~%Warning: Ring ~M not found. Using default.~%" ring))))
266 (t
267 (mtell "~%Ring specification ~M is not recognized. Using default.~%" ring)
268 nil)))
269
270
271;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
272;;
273;; Facilities for evaluating Grobner package expressions
274;; within a prepared environment
275;;
276;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
277
278(defmacro with-monomial-order ((order) &body body)
279 "Evaluate BODY with monomial order set to ORDER."
280 `(let ((*monomial-order* (or (find-order ,order) *monomial-order*)))
281 . ,body))
282
283(defmacro with-coefficient-ring ((ring) &body body)
284 "Evaluate BODY with coefficient ring set to RING."
285 `(let ((+maxima-ring+ (or (find-ring ,ring) +maxima-ring+)))
286 . ,body))
287
288(defmacro with-ring-and-order ((ring order) &body body)
289 "Evaluate BODY with monomial order set to ORDER and coefficient ring set to RING."
290 `(let ((*monomial-order* (or (find-order ,order) *monomial-order*))
291 (+maxima-ring+ (or (find-ring ,ring) +maxima-ring+)))
292 . ,body))
293
294(defmacro with-elimination-orders ((primary secondary elimination-order)
295 &body body)
296 "Evaluate BODY with primary and secondary elimination orders set to PRIMARY and SECONDARY."
297 `(let ((*primary-elimination-order* (or (find-order ,primary) *primary-elimination-order*))
298 (*secondary-elimination-order* (or (find-order ,secondary) *secondary-elimination-order*))
299 (*elimination-order* (or (find-order ,elimination-order) *elimination-order*)))
300 . ,body))
301
302
303;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
304;;
305;; Maxima-level interface functions
306;;
307;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
308
309;; Auxillary function for removing zero polynomial
310(defun remzero (plist) (remove #'poly-zerop plist))
311
312;;Simple operators
313
314(define-binop $poly_add poly-add
315 "Adds two polynomials P and Q")
316
317(define-binop $poly_subtract poly-sub
318 "Subtracts a polynomial Q from P.")
319
320(define-binop $poly_multiply poly-mul
321 "Returns the product of polynomials P and Q.")
322
323(define-binop $poly_s_polynomial spoly
324 "Returns the syzygy polynomial (S-polynomial) of two polynomials P and Q.")
325
326(define-unop $poly_primitive_part poly-primitive-part
327 "Returns the polynomial P divided by GCD of its coefficients.")
328
329(define-unop $poly_normalize poly-normalize
330 "Returns the polynomial P divided by the leading coefficient.")
331
332;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
333;;
334;; Macro facility for writing Maxima-level wrappers for
335;; functions operating on internal representation
336;;
337;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
338
339(defmacro with-parsed-polynomials (((maxima-vars &optional (maxima-new-vars nil new-vars-supplied-p))
340 &key (polynomials nil)
341 (poly-lists nil)
342 (poly-list-lists nil)
343 (value-type nil))
344 &body body
345 &aux (vars (gensym))
346 (new-vars (gensym)))
347 `(let ((,vars (coerce-maxima-list ,maxima-vars))
348 ,@(when new-vars-supplied-p
349 (list `(,new-vars (coerce-maxima-list ,maxima-new-vars)))))
350 (coerce-to-maxima
351 ,value-type
352 (with-coefficient-ring ($poly_coefficient_ring)
353 (with-monomial-order ($poly_monomial_order)
354 (with-elimination-orders ($poly_primary_elimination_order
355 $poly_secondary_elimination_order
356 $poly_elimination_order)
357 (let ,(let ((args nil))
358 (dolist (p polynomials args)
359 (setf args (cons `(,p (parse-poly ,p ,vars)) args)))
360 (dolist (p poly-lists args)
361 (setf args (cons `(,p (parse-poly-list ,p ,vars)) args)))
362 (dolist (p poly-list-lists args)
363 (setf args (cons `(,p (parse-poly-list-list ,p ,vars)) args))))
364 . ,body))))
365 ,(if new-vars-supplied-p
366 `(append ,vars ,new-vars)
367 vars))))
368
369
370;;Functions
371
372(defmfun $poly_expand (p vars)
373 "This function is equivalent to EXPAND(P) if P parses correctly to a polynomial.
374If the representation is not compatible with a polynomial in variables VARS,
375the result is an error."
376 (with-parsed-polynomials ((vars) :polynomials (p)
377 :value-type :polynomial)
378 p))
379
380(defmfun $poly_expt (p n vars)
381 (with-parsed-polynomials ((vars) :polynomials (p) :value-type :polynomial)
382 (poly-expt +maxima-ring+ p n)))
383
384(defmfun $poly_content (p vars)
385 (with-parsed-polynomials ((vars) :polynomials (p))
386 (poly-content +maxima-ring+ p)))
387
388(defmfun $poly_pseudo_divide (f fl vars
389 &aux (vars (coerce-maxima-list vars))
390 (f (parse-poly f vars))
391 (fl (parse-poly-list fl vars)))
392 (multiple-value-bind (quot rem c division-count)
393 (poly-pseudo-divide +maxima-ring+ f fl)
394 `((mlist)
395 ,(coerce-to-maxima :poly-list quot vars)
396 ,(coerce-to-maxima :polynomial rem vars)
397 ,c
398 ,division-count)))
399
400(defmfun $poly_exact_divide (f g vars)
401 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
402 (poly-exact-divide +maxima-ring+ f g)))
403
404(defmfun $poly_normal_form (f fl vars)
405 (with-parsed-polynomials ((vars) :polynomials (f)
406 :poly-lists (fl)
407 :value-type :polynomial)
408 (normal-form +maxima-ring+ f (remzero fl) nil)))
409
410(defmfun $poly_buchberger_criterion (g vars)
411 (with-parsed-polynomials ((vars) :poly-lists (g) :value-type :logical)
412 (buchberger-criterion +maxima-ring+ g)))
413
414(defmfun $poly_buchberger (fl vars)
415 (with-parsed-polynomials ((vars) :poly-lists (fl) :value-type :poly-list)
416 (buchberger +maxima-ring+ (remzero fl) 0 nil)))
417
418(defmfun $poly_reduction (plist vars)
419 (with-parsed-polynomials ((vars) :poly-lists (plist)
420 :value-type :poly-list)
421 (reduction +maxima-ring+ plist)))
422
423(defmfun $poly_minimization (plist vars)
424 (with-parsed-polynomials ((vars) :poly-lists (plist)
425 :value-type :poly-list)
426 (minimization plist)))
427
428(defmfun $poly_normalize_list (plist vars)
429 (with-parsed-polynomials ((vars) :poly-lists (plist)
430 :value-type :poly-list)
431 (poly-normalize-list +maxima-ring+ plist)))
432
433(defmfun $poly_grobner (f vars)
434 (with-parsed-polynomials ((vars) :poly-lists (f)
435 :value-type :poly-list)
436 (grobner +maxima-ring+ (remzero f))))
437
438(defmfun $poly_reduced_grobner (f vars)
439 (with-parsed-polynomials ((vars) :poly-lists (f)
440 :value-type :poly-list)
441 (reduced-grobner +maxima-ring+ (remzero f))))
442
443(defmfun $poly_depends_p (p var mvars
444 &aux (vars (coerce-maxima-list mvars))
445 (pos (position var vars)))
446 (if (null pos)
447 (merror "~%Variable ~M not in the list of variables ~M." var mvars)
448 (poly-depends-p (parse-poly p vars) pos)))
449
450(defmfun $poly_elimination_ideal (flist k vars)
451 (with-parsed-polynomials ((vars) :poly-lists (flist)
452 :value-type :poly-list)
453 (elimination-ideal +maxima-ring+ flist k nil 0)))
454
455(defmfun $poly_colon_ideal (f g vars)
456 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
457 (colon-ideal +maxima-ring+ f g nil)))
458
459(defmfun $poly_ideal_intersection (f g vars)
460 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
461 (ideal-intersection +maxima-ring+ f g nil)))
462
463(defmfun $poly_lcm (f g vars)
464 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
465 (poly-lcm +maxima-ring+ f g)))
466
467(defmfun $poly_gcd (f g vars)
468 ($first ($divide (m* f g) ($poly_lcm f g vars))))
469
470(defmfun $poly_grobner_equal (g1 g2 vars)
471 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
472 (grobner-equal +maxima-ring+ g1 g2)))
473
474(defmfun $poly_grobner_subsetp (g1 g2 vars)
475 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
476 (grobner-subsetp +maxima-ring+ g1 g2)))
477
478(defmfun $poly_grobner_member (p g vars)
479 (with-parsed-polynomials ((vars) :polynomials (p) :poly-lists (g))
480 (grobner-member +maxima-ring+ p g)))
481
482(defmfun $poly_ideal_saturation1 (f p vars)
483 (with-parsed-polynomials ((vars) :poly-lists (f) :polynomials (p)
484 :value-type :poly-list)
485 (ideal-saturation-1 +maxima-ring+ f p 0)))
486
487(defmfun $poly_saturation_extension (f plist vars new-vars)
488 (with-parsed-polynomials ((vars new-vars)
489 :poly-lists (f plist)
490 :value-type :poly-list)
491 (saturation-extension +maxima-ring+ f plist)))
492
493(defmfun $poly_polysaturation_extension (f plist vars new-vars)
494 (with-parsed-polynomials ((vars new-vars)
495 :poly-lists (f plist)
496 :value-type :poly-list)
497 (polysaturation-extension +maxima-ring+ f plist)))
498
499(defmfun $poly_ideal_polysaturation1 (f plist vars)
500 (with-parsed-polynomials ((vars) :poly-lists (f plist)
501 :value-type :poly-list)
502 (ideal-polysaturation-1 +maxima-ring+ f plist 0 nil)))
503
504(defmfun $poly_ideal_saturation (f g vars)
505 (with-parsed-polynomials ((vars) :poly-lists (f g)
506 :value-type :poly-list)
507 (ideal-saturation +maxima-ring+ f g 0 nil)))
508
509(defmfun $poly_ideal_polysaturation (f ideal-list vars)
510 (with-parsed-polynomials ((vars) :poly-lists (f)
511 :poly-list-lists (ideal-list)
512 :value-type :poly-list)
513 (ideal-polysaturation +maxima-ring+ f ideal-list 0 nil)))
514
515(defmfun $poly_lt (f vars)
516 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
517 (make-poly-from-termlist (list (poly-lt f)))))
518
519(defmfun $poly_lm (f vars)
520 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
521 (make-poly-from-termlist (list (make-term :monom (poly-lm f) :coeff (funcall (ring-unit ring)))))))
522
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