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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;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
122;;
123;; Order utilities
124;;
125;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
126
127(defun find-ring (ring)
128 "This function returns the ring structure bases on input symbol."
129 (cond
130 ((null ring) nil)
131 ((symbolp ring)
132 (case ring
133 ((maxima-ring :maxima-ring #:maxima-ring $expression_ring #:expression_ring)
134 *maxima-ring*)
135 ((ring-of-integers :ring-of-integers #:ring-of-integers $ring_of_integers) *ring-of-integers*)
136 (otherwise
137 (mtell "~%Warning: Ring ~M not found. Using default.~%" ring))))
138 (t
139 (mtell "~%Ring specification ~M is not recognized. Using default.~%" ring)
140 nil)))
141
142(defun find-order (order)
143 "This function returns the order function bases on its name."
144 (cond
145 ((null order) nil)
146 ((symbolp order)
147 (case order
148 ((lex :lex $lex #:lex)
149 #'lex>)
150 ((grlex :grlex $grlex #:grlex)
151 #'grlex>)
152 ((grevlex :grevlex $grevlex #:grevlex)
153 #'grevlex>)
154 ((invlex :invlex $invlex #:invlex)
155 #'invlex>)
156 (otherwise
157 (mtell "~%Warning: Order ~M not found. Using default.~%" order))))
158 (t
159 (mtell "~%Order specification ~M is not recognized. Using default.~%" order)
160 nil)))
161
162(defun maxima->poly (expr vars
163 &optional
164 (ring $poly_coefficient_ring)
165 (order $poly_monomial_order)
166 (primary-elimination-order $poly_primary_elimination_order)
167 (secondary-elimination-order $poly_secondary_elimination_order)
168 &aux
169 (vars (coerce-maxima-list vars))
170 (ring-and-order (make-ring-and-order
171 :ring (find-ring)
172 :order (find-order)
173 :primary-elimination-order primary-elimination-order
174 :secondary-elimination-order secondary-elimination-order))
175 (ring (ro-order ring-and-order)))
176 "Convert a maxima polynomial expression EXPR in variables VARS to internal form."
177 (labels ((parse (arg) (maxima->poly arg vars))
178 (parse-list (args) (mapcar #'parse args)))
179 (cond
180 ((eql expr 0) (make-poly-zero))
181 ((member expr vars :test #'equal-test-p)
182 (let ((pos (position expr vars :test #'equal-test-p)))
183 (make-poly-variable ring (length vars) pos)))
184 ((free-of-vars expr vars)
185 ;;This means that variable-free CRE and Poisson forms will be converted
186 ;;to coefficients intact
187 (coerce-coeff ring expr vars))
188 (t
189 (case (caar expr)
190 (mplus (reduce #'(lambda (x y) (poly-add ring-and-order x y)) (parse-list (cdr expr))))
191 (mminus (poly-uminus ring (parse (cadr expr))))
192 (mtimes
193 (if (endp (cddr expr)) ;unary
194 (parse (cdr expr))
195 (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (parse-list (cdr expr)))))
196 (mexpt
197 (cond
198 ((member (cadr expr) vars :test #'equal-test-p)
199 ;;Special handling of (expt var pow)
200 (let ((pos (position (cadr expr) vars :test #'equal-test-p)))
201 (make-poly-variable ring (length vars) pos (caddr expr))))
202 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
203 ;; Negative power means division in coefficient ring
204 ;; Non-integer power means non-polynomial coefficient
205 (mtell "~%Warning: Expression ~%~M~%contains power which is not a positive integer. Parsing as coefficient.~%"
206 expr)
207 (coerce-coeff ring expr vars))
208 (t (poly-expt ring (parse (cadr expr)) (caddr expr)))))
209 (mrat (parse ($ratdisrep expr)))
210 (mpois (parse ($outofpois expr)))
211 (otherwise
212 (coerce-coeff ring expr vars)))))))
213
214#|
215
216(defun parse-poly-list (expr vars)
217 "Parse a Maxima representation of a list of polynomials."
218 (case (caar expr)
219 (mlist (mapcar #'(lambda (p) (parse-poly p vars)) (cdr expr)))
220 (t (merror "Expression ~M is not a list of polynomials in variables ~M."
221 expr vars))))
222
223(defun parse-poly-list-list (poly-list-list vars)
224 "Parse a Maxima representation of a list of lists of polynomials."
225 (mapcar #'(lambda (g) (parse-poly-list g vars)) (coerce-maxima-list poly-list-list)))
226
227
228;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
229;;
230;; Conversion from internal form to Maxima general form
231;;
232;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
233
234(defun maxima-head ()
235 (if $poly_return_term_list
236 '(mlist)
237 '(mplus)))
238
239(defun coerce-to-maxima (poly-type object vars)
240 (case poly-type
241 (:polynomial
242 `(,(maxima-head) ,@(mapcar #'(lambda (term) (coerce-to-maxima :term term vars)) (poly-termlist object))))
243 (:poly-list
244 `((mlist) ,@(mapcar #'(lambda (p) (funcall *ratdisrep-fun* (coerce-to-maxima :polynomial p vars))) object)))
245 (:term
246 `((mtimes) ,(funcall *ratdisrep-fun* (term-coeff object))
247 ,@(mapcar #'(lambda (var power) `((mexpt) ,var ,power))
248 vars (coerce (term-monom object) 'list))))
249 ;; Assumes that Lisp and Maxima logicals coincide
250 (:logical object)
251 (otherwise
252 object)))
253
254
255;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
256;;
257;; Unary and binary operation definition facility
258;;
259;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
260
261(defmacro define-unop (maxima-name fun-name
262 &optional (documentation nil documentation-supplied-p))
263 "Define a MAXIMA-level unary operator MAXIMA-NAME corresponding to unary function FUN-NAME."
264 `(defun ,maxima-name (p vars
265 &aux
266 (vars (coerce-maxima-list vars))
267 (p (parse-poly p vars)))
268 ,@(when documentation-supplied-p (list documentation))
269 (coerce-to-maxima :polynomial (,fun-name *maxima-ring* p) vars)))
270
271(defmacro define-binop (maxima-name fun-name
272 &optional (documentation nil documentation-supplied-p))
273 "Define a MAXIMA-level binary operator MAXIMA-NAME corresponding to binary function FUN-NAME."
274 `(defmfun ,maxima-name (p q vars
275 &aux
276 (vars (coerce-maxima-list vars))
277 (p (parse-poly p vars))
278 (q (parse-poly q vars)))
279 ,@(when documentation-supplied-p (list documentation))
280 (coerce-to-maxima :polynomial (,fun-name *maxima-ring* p q) vars)))
281
282
283;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
284;;
285;; Facilities for evaluating Grobner package expressions
286;; within a prepared environment
287;;
288;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
289
290(defmacro with-monomial-order ((order) &body body)
291 "Evaluate BODY with monomial order set to ORDER."
292 `(let ((*monomial-order* (or (find-order ,order) *monomial-order*)))
293 . ,body))
294
295(defmacro with-coefficient-ring ((ring) &body body)
296 "Evaluate BODY with coefficient ring set to RING."
297 `(let ((*maxima-ring* (or (find-ring ,ring) *maxima-ring*)))
298 . ,body))
299
300(defmacro with-ring-and-order ((ring order) &body body)
301 "Evaluate BODY with monomial order set to ORDER and coefficient ring set to RING."
302 `(let ((*monomial-order* (or (find-order ,order) *monomial-order*))
303 (*maxima-ring* (or (find-ring ,ring) *maxima-ring*)))
304 . ,body))
305
306(defmacro with-elimination-orders ((primary secondary elimination-order)
307 &body body)
308 "Evaluate BODY with primary and secondary elimination orders set to PRIMARY and SECONDARY."
309 `(let ((*primary-elimination-order* (or (find-order ,primary) *primary-elimination-order*))
310 (*secondary-elimination-order* (or (find-order ,secondary) *secondary-elimination-order*))
311 (*elimination-order* (or (find-order ,elimination-order) *elimination-order*)))
312 . ,body))
313
314
315;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
316;;
317;; Maxima-level interface functions
318;;
319;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
320
321;; Auxillary function for removing zero polynomial
322(defun remzero (plist) (remove #'poly-zerop plist))
323
324;;Simple operators
325
326(define-binop $poly_add poly-add
327 "Adds two polynomials P and Q")
328
329(define-binop $poly_subtract poly-sub
330 "Subtracts a polynomial Q from P.")
331
332(define-binop $poly_multiply poly-mul
333 "Returns the product of polynomials P and Q.")
334
335(define-binop $poly_s_polynomial spoly
336 "Returns the syzygy polynomial (S-polynomial) of two polynomials P and Q.")
337
338(define-unop $poly_primitive_part poly-primitive-part
339 "Returns the polynomial P divided by GCD of its coefficients.")
340
341(define-unop $poly_normalize poly-normalize
342 "Returns the polynomial P divided by the leading coefficient.")
343
344;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
345;;
346;; Macro facility for writing Maxima-level wrappers for
347;; functions operating on internal representation
348;;
349;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
350
351(defmacro with-parsed-polynomials (((maxima-vars &optional (maxima-new-vars nil new-vars-supplied-p))
352 &key (polynomials nil)
353 (poly-lists nil)
354 (poly-list-lists nil)
355 (value-type nil))
356 &body body
357 &aux (vars (gensym))
358 (new-vars (gensym)))
359 `(let ((,vars (coerce-maxima-list ,maxima-vars))
360 ,@(when new-vars-supplied-p
361 (list `(,new-vars (coerce-maxima-list ,maxima-new-vars)))))
362 (coerce-to-maxima
363 ,value-type
364 (with-coefficient-ring ($poly_coefficient_ring)
365 (with-monomial-order ($poly_monomial_order)
366 (with-elimination-orders ($poly_primary_elimination_order
367 $poly_secondary_elimination_order
368 $poly_elimination_order)
369 (let ,(let ((args nil))
370 (dolist (p polynomials args)
371 (setf args (cons `(,p (parse-poly ,p ,vars)) args)))
372 (dolist (p poly-lists args)
373 (setf args (cons `(,p (parse-poly-list ,p ,vars)) args)))
374 (dolist (p poly-list-lists args)
375 (setf args (cons `(,p (parse-poly-list-list ,p ,vars)) args))))
376 . ,body))))
377 ,(if new-vars-supplied-p
378 `(append ,vars ,new-vars)
379 vars))))
380
381
382;;Functions
383
384(defmfun $poly_expand (p vars)
385 "This function is equivalent to EXPAND(P) if P parses correctly to a polynomial.
386If the representation is not compatible with a polynomial in variables VARS,
387the result is an error."
388 (with-parsed-polynomials ((vars) :polynomials (p)
389 :value-type :polynomial)
390 p))
391
392(defmfun $poly_expt (p n vars)
393 (with-parsed-polynomials ((vars) :polynomials (p) :value-type :polynomial)
394 (poly-expt *maxima-ring* p n)))
395
396(defmfun $poly_content (p vars)
397 (with-parsed-polynomials ((vars) :polynomials (p))
398 (poly-content *maxima-ring* p)))
399
400(defmfun $poly_pseudo_divide (f fl vars
401 &aux (vars (coerce-maxima-list vars))
402 (f (parse-poly f vars))
403 (fl (parse-poly-list fl vars)))
404 (multiple-value-bind (quot rem c division-count)
405 (poly-pseudo-divide *maxima-ring* f fl)
406 `((mlist)
407 ,(coerce-to-maxima :poly-list quot vars)
408 ,(coerce-to-maxima :polynomial rem vars)
409 ,c
410 ,division-count)))
411
412(defmfun $poly_exact_divide (f g vars)
413 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
414 (poly-exact-divide *maxima-ring* f g)))
415
416(defmfun $poly_normal_form (f fl vars)
417 (with-parsed-polynomials ((vars) :polynomials (f)
418 :poly-lists (fl)
419 :value-type :polynomial)
420 (normal-form *maxima-ring* f (remzero fl) nil)))
421
422(defmfun $poly_buchberger_criterion (g vars)
423 (with-parsed-polynomials ((vars) :poly-lists (g) :value-type :logical)
424 (buchberger-criterion *maxima-ring* g)))
425
426(defmfun $poly_buchberger (fl vars)
427 (with-parsed-polynomials ((vars) :poly-lists (fl) :value-type :poly-list)
428 (buchberger *maxima-ring* (remzero fl) 0 nil)))
429
430(defmfun $poly_reduction (plist vars)
431 (with-parsed-polynomials ((vars) :poly-lists (plist)
432 :value-type :poly-list)
433 (reduction *maxima-ring* plist)))
434
435(defmfun $poly_minimization (plist vars)
436 (with-parsed-polynomials ((vars) :poly-lists (plist)
437 :value-type :poly-list)
438 (minimization plist)))
439
440(defmfun $poly_normalize_list (plist vars)
441 (with-parsed-polynomials ((vars) :poly-lists (plist)
442 :value-type :poly-list)
443 (poly-normalize-list *maxima-ring* plist)))
444
445(defmfun $poly_grobner (f vars)
446 (with-parsed-polynomials ((vars) :poly-lists (f)
447 :value-type :poly-list)
448 (grobner *maxima-ring* (remzero f))))
449
450(defmfun $poly_reduced_grobner (f vars)
451 (with-parsed-polynomials ((vars) :poly-lists (f)
452 :value-type :poly-list)
453 (reduced-grobner *maxima-ring* (remzero f))))
454
455(defmfun $poly_depends_p (p var mvars
456 &aux (vars (coerce-maxima-list mvars))
457 (pos (position var vars)))
458 (if (null pos)
459 (merror "~%Variable ~M not in the list of variables ~M." var mvars)
460 (poly-depends-p (parse-poly p vars) pos)))
461
462(defmfun $poly_elimination_ideal (flist k vars)
463 (with-parsed-polynomials ((vars) :poly-lists (flist)
464 :value-type :poly-list)
465 (elimination-ideal *maxima-ring* flist k nil 0)))
466
467(defmfun $poly_colon_ideal (f g vars)
468 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
469 (colon-ideal *maxima-ring* f g nil)))
470
471(defmfun $poly_ideal_intersection (f g vars)
472 (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
473 (ideal-intersection *maxima-ring* f g nil)))
474
475(defmfun $poly_lcm (f g vars)
476 (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
477 (poly-lcm *maxima-ring* f g)))
478
479(defmfun $poly_gcd (f g vars)
480 ($first ($divide (m* f g) ($poly_lcm f g vars))))
481
482(defmfun $poly_grobner_equal (g1 g2 vars)
483 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
484 (grobner-equal *maxima-ring* g1 g2)))
485
486(defmfun $poly_grobner_subsetp (g1 g2 vars)
487 (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
488 (grobner-subsetp *maxima-ring* g1 g2)))
489
490(defmfun $poly_grobner_member (p g vars)
491 (with-parsed-polynomials ((vars) :polynomials (p) :poly-lists (g))
492 (grobner-member *maxima-ring* p g)))
493
494(defmfun $poly_ideal_saturation1 (f p vars)
495 (with-parsed-polynomials ((vars) :poly-lists (f) :polynomials (p)
496 :value-type :poly-list)
497 (ideal-saturation-1 *maxima-ring* f p 0)))
498
499(defmfun $poly_saturation_extension (f plist vars new-vars)
500 (with-parsed-polynomials ((vars new-vars)
501 :poly-lists (f plist)
502 :value-type :poly-list)
503 (saturation-extension *maxima-ring* f plist)))
504
505(defmfun $poly_polysaturation_extension (f plist vars new-vars)
506 (with-parsed-polynomials ((vars new-vars)
507 :poly-lists (f plist)
508 :value-type :poly-list)
509 (polysaturation-extension *maxima-ring* f plist)))
510
511(defmfun $poly_ideal_polysaturation1 (f plist vars)
512 (with-parsed-polynomials ((vars) :poly-lists (f plist)
513 :value-type :poly-list)
514 (ideal-polysaturation-1 *maxima-ring* f plist 0 nil)))
515
516(defmfun $poly_ideal_saturation (f g vars)
517 (with-parsed-polynomials ((vars) :poly-lists (f g)
518 :value-type :poly-list)
519 (ideal-saturation *maxima-ring* f g 0 nil)))
520
521(defmfun $poly_ideal_polysaturation (f ideal-list vars)
522 (with-parsed-polynomials ((vars) :poly-lists (f)
523 :poly-list-lists (ideal-list)
524 :value-type :poly-list)
525 (ideal-polysaturation *maxima-ring* f ideal-list 0 nil)))
526
527(defmfun $poly_lt (f vars)
528 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
529 (make-poly-from-termlist (list (poly-lt f)))))
530
531(defmfun $poly_lm (f vars)
532 (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
533 (make-poly-from-termlist (list (make-term (poly-lm f) (funcall (ring-unit *maxima-ring*)))))))
534
535|#
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