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1<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997)
2 by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
3* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
4* with significant contributions from:
5 Jens Lippman, Marek Rouchal, Martin Wilck and others -->
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8<TITLE>The Polynomial Package</TITLE>
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34<BR>
35<B> Next:</B> <A NAME="tex2html1035"
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38 HREF="manual.html">CGBLisp User Guide and</A>
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40 HREF="node9.html">The Monomial Order Package</A>
41<BR>
42<BR>
43<!--End of Navigation Panel-->
44<!--Table of Child-Links-->
45<A NAME="CHILD_LINKS"><strong>Subsections</strong></A>
46<UL>
47<LI><A NAME="tex2html1036"
48 HREF="node10.html#SECTION000100010000000000000">
49<I>scalar<MATH CLASS="INLINE">
50-
51</MATH>times<MATH CLASS="INLINE">
52-
53</MATH>poly</I></A>
54<LI><A NAME="tex2html1037"
55 HREF="node10.html#SECTION000100020000000000000">
56<I>term<MATH CLASS="INLINE">
57-
58</MATH>times<MATH CLASS="INLINE">
59-
60</MATH>poly</I></A>
61<LI><A NAME="tex2html1038"
62 HREF="node10.html#SECTION000100030000000000000">
63<I>monom<MATH CLASS="INLINE">
64-
65</MATH>times<MATH CLASS="INLINE">
66-
67</MATH>poly</I></A>
68<LI><A NAME="tex2html1039"
69 HREF="node10.html#SECTION000100040000000000000">
70<I>minus<MATH CLASS="INLINE">
71-
72</MATH>poly</I></A>
73<LI><A NAME="tex2html1040"
74 HREF="node10.html#SECTION000100050000000000000">
75<I>sort<MATH CLASS="INLINE">
76-
77</MATH>poly</I></A>
78<LI><A NAME="tex2html1041"
79 HREF="node10.html#SECTION000100060000000000000">
80<I>poly+</I></A>
81<LI><A NAME="tex2html1042"
82 HREF="node10.html#SECTION000100070000000000000">
83<I>poly<MATH CLASS="INLINE">
84-
85</MATH></I></A>
86<LI><A NAME="tex2html1043"
87 HREF="node10.html#SECTION000100080000000000000">
88<I>poly*</I></A>
89<LI><A NAME="tex2html1044"
90 HREF="node10.html#SECTION000100090000000000000">
91<I>poly<MATH CLASS="INLINE">
92-
93</MATH>op</I></A>
94<LI><A NAME="tex2html1045"
95 HREF="node10.html#SECTION0001000100000000000000">
96<I>poly<MATH CLASS="INLINE">
97-
98</MATH>expt</I></A>
99<LI><A NAME="tex2html1046"
100 HREF="node10.html#SECTION0001000110000000000000">
101<I>poly<MATH CLASS="INLINE">
102-
103</MATH>mexpt</I></A>
104<LI><A NAME="tex2html1047"
105 HREF="node10.html#SECTION0001000120000000000000">
106<I>poly<MATH CLASS="INLINE">
107-
108</MATH>constant<MATH CLASS="INLINE">
109-
110</MATH>p</I></A>
111<LI><A NAME="tex2html1048"
112 HREF="node10.html#SECTION0001000130000000000000">
113<I>poly<MATH CLASS="INLINE">
114-
115</MATH>extend</I></A>
116<LI><A NAME="tex2html1049"
117 HREF="node10.html#SECTION0001000140000000000000">
118<I>poly<MATH CLASS="INLINE">
119-
120</MATH>extend<MATH CLASS="INLINE">
121-
122</MATH>end</I></A>
123<LI><A NAME="tex2html1050"
124 HREF="node10.html#SECTION0001000150000000000000">
125<I>poly<MATH CLASS="INLINE">
126-
127</MATH>zerop</I></A>
128<LI><A NAME="tex2html1051"
129 HREF="node10.html#SECTION0001000160000000000000">
130<I>lt</I></A>
131<LI><A NAME="tex2html1052"
132 HREF="node10.html#SECTION0001000170000000000000">
133<I>lm</I></A>
134<LI><A NAME="tex2html1053"
135 HREF="node10.html#SECTION0001000180000000000000">
136<I>lc</I></A>
137</UL>
138<!--End of Table of Child-Links-->
139<HR>
140<H1><A NAME="SECTION000100000000000000000">
141The Polynomial Package</A>
142</H1>
143<H4><A NAME="SECTION000100010000000000000">
144<I>scalar<MATH CLASS="INLINE">
145-
146</MATH>times<MATH CLASS="INLINE">
147-
148</MATH>poly</I></A>
149</H4>
150<P><IMG WIDTH="481" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
151 SRC="img153.gif"
152 ALT="$\textstyle\parbox{\pboxargslen}{\em c p {\sf \&optional} (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
153<BLOCKQUOTE>
154Return product of a scalar C by a polynomial P with coefficient ring
155RING. </BLOCKQUOTE><H4><A NAME="SECTION000100020000000000000">
156<I>term<MATH CLASS="INLINE">
157-
158</MATH>times<MATH CLASS="INLINE">
159-
160</MATH>poly</I></A>
161</H4>
162<P><IMG WIDTH="488" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
163 SRC="img154.gif"
164 ALT="$\textstyle\parbox{\pboxargslen}{\em term f {\sf \&optional} (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
165<BLOCKQUOTE>
166Return product of a term TERM by a polynomial F with coefficient ring
167RING. </BLOCKQUOTE><H4><A NAME="SECTION000100030000000000000">
168<I>monom<MATH CLASS="INLINE">
169-
170</MATH>times<MATH CLASS="INLINE">
171-
172</MATH>poly</I></A>
173</H4>
174<P><IMG WIDTH="453" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
175 SRC="img129.gif"
176 ALT="$\textstyle\parbox{\pboxargslen}{\em m f \/}$"> [<EM>FUNCTION</EM>]
177<BLOCKQUOTE>
178Return product of a monomial M by a polynomial F with coefficient
179ring RING. </BLOCKQUOTE><H4><A NAME="SECTION000100040000000000000">
180<I>minus<MATH CLASS="INLINE">
181-
182</MATH>poly</I></A>
183</H4>
184<P><IMG WIDTH="529" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
185 SRC="img155.gif"
186 ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
187<BLOCKQUOTE>
188Changes the sign of a polynomial F with coefficients in coefficient
189ring RING, and returns the result.</BLOCKQUOTE><H4><A NAME="SECTION000100050000000000000">
190<I>sort<MATH CLASS="INLINE">
191-
192</MATH>poly</I></A>
193</H4>
194<P><IMG WIDTH="544" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
195 SRC="img156.gif"
196 ALT="$\textstyle\parbox{\pboxargslen}{\em poly {\sf \&optional} (pred
197 \char93 'lex$\gt$) (start
198 0) (end
199 (unless
200 (null poly)
201 (length
202 (caar poly)))) \/}$"> [<EM>FUNCTION</EM>]
203<BLOCKQUOTE>
204Destructively Sorts a polynomial POLY by predicate PRED; the
205predicate is assumed to take arguments START and END in addition to
206the pair of monomials, as the functions in the ORDER package do.</BLOCKQUOTE><H4><A NAME="SECTION000100060000000000000">
207<I>poly+</I></A>
208</H4>
209<P><IMG WIDTH="570" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
210 SRC="img157.gif"
211 ALT="$\textstyle\parbox{\pboxargslen}{\em p q {\sf \&optional} (pred \char93 'lex$\gt$) (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
212<BLOCKQUOTE>
213Returns the sum of two polynomials P and Q with coefficients in
214ring RING, with terms ordered according to monomial order PRED.</BLOCKQUOTE><H4><A NAME="SECTION000100070000000000000">
215<I>poly<MATH CLASS="INLINE">
216-
217</MATH></I></A>
218</H4>
219<P><IMG WIDTH="570" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
220 SRC="img157.gif"
221 ALT="$\textstyle\parbox{\pboxargslen}{\em p q {\sf \&optional} (pred \char93 'lex$\gt$) (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
222<BLOCKQUOTE>
223Returns the difference of two polynomials P and Q with coefficients
224in ring RING, with terms ordered according to monomial order PRED.</BLOCKQUOTE><H4><A NAME="SECTION000100080000000000000">
225<I>poly*</I></A>
226</H4>
227<P><IMG WIDTH="570" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
228 SRC="img157.gif"
229 ALT="$\textstyle\parbox{\pboxargslen}{\em p q {\sf \&optional} (pred \char93 'lex$\gt$) (ring *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
230<BLOCKQUOTE>
231Returns the product of two polynomials P and Q with coefficients in
232ring RING, with terms ordered according to monomial order PRED.</BLOCKQUOTE><H4><A NAME="SECTION000100090000000000000">
233<I>poly<MATH CLASS="INLINE">
234-
235</MATH>op</I></A>
236</H4>
237<P><IMG WIDTH="553" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
238 SRC="img158.gif"
239 ALT="$\textstyle\parbox{\pboxargslen}{\em f m g pred ring \/}$"> [<EM>FUNCTION</EM>]
240<BLOCKQUOTE>
241Returns F<MATH CLASS="INLINE">
242-
243</MATH>M*G, where F and G are polynomials with coefficients in
244ring RING, ordered according to monomial order PRED and M is a
245monomial.</BLOCKQUOTE><H4><A NAME="SECTION0001000100000000000000">
246<I>poly<MATH CLASS="INLINE">
247-
248</MATH>expt</I></A>
249</H4>
250<P><IMG WIDTH="540" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
251 SRC="img159.gif"
252 ALT="$\textstyle\parbox{\pboxargslen}{\em poly n {\sf \&optional} (pred
253 \char93 'lex$\gt$) (ring
254 *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
255<BLOCKQUOTE>
256Exponentiate a polynomial POLY to power N. The terms of the
257polynomial are assumed to be ordered by monomial order PRED and with
258coefficients in ring RING. Use the Chinese algorithm; assume N<MATH CLASS="INLINE">
259&gt;
260</MATH>=0
261and POLY is non<MATH CLASS="INLINE">
262-
263</MATH>zero (not NIL).</BLOCKQUOTE><H4><A NAME="SECTION0001000110000000000000">
264<I>poly<MATH CLASS="INLINE">
265-
266</MATH>mexpt</I></A>
267</H4>
268<P><IMG WIDTH="527" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
269 SRC="img160.gif"
270 ALT="$\textstyle\parbox{\pboxargslen}{\em plist monom {\sf \&optional} (pred
271 \char93 'lex$\gt$) (ring
272 *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
273<BLOCKQUOTE>
274Raise a polynomial vector represented ad a list of polynomials
275PLIST to power MULTIINDEX. Every polynomial has its terms ordered by
276predicate PRED and coefficients in the ring RING.</BLOCKQUOTE><H4><A NAME="SECTION0001000120000000000000">
277<I>poly<MATH CLASS="INLINE">
278-
279</MATH>constant<MATH CLASS="INLINE">
280-
281</MATH>p</I></A>
282</H4>
283<P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
284 SRC="img80.gif"
285 ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
286<BLOCKQUOTE>
287Returns T if P is a constant polynomial.</BLOCKQUOTE><H4><A NAME="SECTION0001000130000000000000">
288<I>poly<MATH CLASS="INLINE">
289-
290</MATH>extend</I></A>
291</H4>
292<P><IMG WIDTH="524" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
293 SRC="img161.gif"
294 ALT="$\textstyle\parbox{\pboxargslen}{\em p {\sf \&optional} (m (list 0)) \/}$"> [<EM>FUNCTION</EM>]
295<BLOCKQUOTE>
296Given a polynomial P in k[x[r+1],...,xn], it returns the same
297polynomial as an element of k[x1,...,xn], optionally multiplying it
298by a monomial x1&#94;m1*x2&#94;m2*...*xr&#94;mr,
299where m=(m1,m2,...,mr) is a multiindex.</BLOCKQUOTE><H4><A NAME="SECTION0001000140000000000000">
300<I>poly<MATH CLASS="INLINE">
301-
302</MATH>extend<MATH CLASS="INLINE">
303-
304</MATH>end</I></A>
305</H4>
306<P><IMG WIDTH="524" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
307 SRC="img161.gif"
308 ALT="$\textstyle\parbox{\pboxargslen}{\em p {\sf \&optional} (m (list 0)) \/}$"> [<EM>FUNCTION</EM>]
309<BLOCKQUOTE>
310Similar to POLY<MATH CLASS="INLINE">
311-
312</MATH>EXTEND, but it adds new variables at the end.</BLOCKQUOTE><H4><A NAME="SECTION0001000150000000000000">
313<I>poly<MATH CLASS="INLINE">
314-
315</MATH>zerop</I></A>
316</H4>
317<P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
318 SRC="img80.gif"
319 ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
320<BLOCKQUOTE>
321Returns T if P is a zero polynomial.</BLOCKQUOTE><H4><A NAME="SECTION0001000160000000000000">
322<I>lt</I></A>
323</H4>
324<P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
325 SRC="img80.gif"
326 ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
327<BLOCKQUOTE>
328Returns the leading term of a polynomial P.</BLOCKQUOTE><H4><A NAME="SECTION0001000170000000000000">
329<I>lm</I></A>
330</H4>
331<P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
332 SRC="img80.gif"
333 ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
334<BLOCKQUOTE>
335Returns the leading monomial of a polynomial P.</BLOCKQUOTE><H4><A NAME="SECTION0001000180000000000000">
336<I>lc</I></A>
337</H4>
338<P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
339 SRC="img80.gif"
340 ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
341<BLOCKQUOTE>
342Returns the leading coefficient of a polynomial P.</BLOCKQUOTE><HR>
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364<ADDRESS>
365<I>Marek Rychlik</I>
366<BR><I>3/21/1998</I>
367</ADDRESS>
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