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source: branches/f4grobner/mx-grobner.lisp@ 1645

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