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31<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
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33<H1>Formatted manual of bfct.lib</H1></P><P>
34
35This file contains the formatted documentation of bfct.lib
36</P><P>
37
38<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0> 
39<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC1">1. Singular libraries</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
40<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC14">2. Index</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
41</TABLE></BLOCKQUOTE>
42<P>
43
44<HR SIZE=1>
45<A NAME="SEC1"></A>
46<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
47<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Top"> &lt; </A>]</TD>
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53<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
54<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
55<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
56</TR></TABLE>
57<A NAME="Singular libraries"></A>
58<H1> 1. Singular libraries </H1>
59<!--docid::SEC1::-->
60<P>
61
62<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0> 
63<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC2">1.1 bfct_lib</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP"></TD></TR>
64</TABLE></BLOCKQUOTE>
65 
66<A NAME="bfct_lib"></A>
67<HR SIZE="6">
68<A NAME="SEC2"></A>
69<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
70<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC1"> &lt; </A>]</TD>
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72<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[ &lt;&lt; ]</TD>
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76<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
77<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
78<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
79</TR></TABLE>
80<H2> 1.1 bfct_lib </H2>
81<!--docid::SEC2::-->
82<P>
83
84<TABLE><tr><td>&nbsp;</td><td class=example><pre>-------BEGIN OF PART WHICH IS INCLUDED IN MANUAL-----
85</pre></td></tr></table></P><P>
86
87<A NAME="IDX1"></A>
88<A NAME="IDX2"></A>
89<DL COMPACT>
90<DT><STRONG>Library:</STRONG>
91<DD>bfct.lib
92<DT><STRONG>Purpose:</STRONG>
93<DD>     M. Noro's Algorithm for Bernstein-Sato polynomial
94<DT><STRONG>Authors:</STRONG>
95<DD>Daniel Andres, daniel.andres@math.rwth-aachen.de
96<BR> Viktor Levandovskyy, levandov@math.rwth-aachen.de
97<P>
98
99<DT><STRONG>Theory:</STRONG>
100<DD>Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R,
101<BR> one is interested in the global Bernstein-Sato polynomial b(s) in K[s],
102<BR> defined to be the monic polynomial, satisfying a functional identity
103<BR> L * f^{s+1} = b(s) f^s, for some operator L in D[s].
104<BR> Here, D stands for an n-th Weyl algebra K&#60;x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1&#62;
105<BR> One is interested in the following data:
106<BR> global Bernstein-Sato polynomial in K[s] and
107<BR> the list of all roots of b(s), which are known to be rational, with their multiplicities.
108<P>
109
110</DL>
111<P>
112
113<STRONG>Main procedures:</STRONG>
114<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0> 
115<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC3">1.1.0.1 bfct</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the global Bernstein-Sato polynomial of a given poly</TD></TR>
116<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC4">1.1.0.2 bfctsyz</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the global Bernstein-Sato polynomial of a given poly</TD></TR>
117<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC5">1.1.0.3 bfctonestep</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the global Bernstein-Sato polynomial of a given poly</TD></TR>
118<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC6">1.1.0.4 bfctideal</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the global b-function of a given ideal w.r.t. a given weight</TD></TR>
119<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC7">1.1.0.5 minpol</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the minimal polynomial of the endormorphism in basering modulo ideal given by a poly</TD></TR>
120<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC8">1.1.0.6 minpolsyz</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the minimal polynomial of the endormorphism in basering modulo ideal given by a poly</TD></TR>
121<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC9">1.1.0.7 linreduce</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">reduce a poly by linear reductions of its leading term</TD></TR>
122<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC10">1.1.0.8 ncsolve</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">find and compute a linear dependency of the elements of an ideal</TD></TR>
123</TABLE></BLOCKQUOTE>
124<STRONG>Auxiliary procedures:</STRONG>
125<BLOCKQUOTE><TABLE BORDER=0 CELLSPACING=0> 
126<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC11">1.1.0.9 ispositive</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">check whether all entries of an intvec are positive</TD></TR>
127<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC12">1.1.0.10 isin</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">check whether an element is a member of a list</TD></TR>
128<TR><TD ALIGN="left" VALIGN="TOP"><A HREF="bfct.html#SEC13">1.1.0.11 scalarprod</A></TD><TD>&nbsp;&nbsp;</TD><TD ALIGN="left" VALIGN="TOP">compute the standard scalar product of two intvecs</TD></TR>
129</TABLE></BLOCKQUOTE>
130<STRONG>See also:</STRONG>
131@xref{dmod_lib};
132@xref{dmodapp_lib};
133@xref{gmssing_lib}.
134<P>
135
136<A NAME="bfct"></A>
137<HR SIZE="6">
138<A NAME="SEC3"></A>
139<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
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142<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[ &lt;&lt; ]</TD>
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144<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14"> &gt;&gt; </A>]</TD>
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146<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
147<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
148<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
149</TR></TABLE>
150<H4> 1.1.0.1 bfct </H4>
151<!--docid::SEC3::-->
152Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
153<P>
154
155<DL COMPACT>
156<DT><STRONG>Usage:</STRONG>
157<DD>bfct(f [,s,t,v]); f a poly, s,t optional ints, v an optional intvec
158<P>
159
160<DT><STRONG>Return:</STRONG>
161<DD>list of roots of the Bernstein-Sato polynomial bs(f) and their multiplicies
162<P>
163
164<DT><STRONG>Purpose:</STRONG>
165<DD>compute the global Bernstein-Sato polynomial for a hypersurface, defined by f, according to the algorithm by Masayuki Noro
166<P>
167
168<DT><STRONG>Note:</STRONG>
169<DD>In this proc, a system of linear equations is solved by linear reductions.
170<BR> If s&#60;&#62;0, <CODE>std</CODE> is used for Groebner basis computations,
171<BR> otherwise, and by default, <CODE>slimgb</CODE> is used.
172<BR> If t&#60;&#62;0, a matrix ordering is used for Groebner basis computations,
173<BR> otherwise, and by default, a block ordering is used.
174<BR> If v is a positive weight vector, v is used for homogenization computations,
175<BR> otherwise and by default, no weights are used.
176<BR> If printlevel=1, progress debug messages will be printed,
177<BR> if printlevel&#62;=2, all the debug messages will be printed.
178<P>
179
180</DL>
181<STRONG>Example:</STRONG>
182<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
183ring r = 0,(x,y),dp;
184poly f = x^2+y^3+x*y^2;
185bfct(f);
186==> [1]:
187==>    _[1]=-7/6
188==>    _[2]=-1
189==>    _[3]=-5/6
190==> [2]:
191==>    1,1,1
192intvec v = 3,2;
193bfct(f,1,0,v);
194==> [1]:
195==>    _[1]=-7/6
196==>    _[2]=-1
197==>    _[3]=-5/6
198==> [2]:
199==>    1,1,1
200</FONT></pre></td></tr></table><P>
201
202<A NAME="bfctsyz"></A>
203<HR SIZE="6">
204<A NAME="SEC4"></A>
205<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
206<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC3"> &lt; </A>]</TD>
207<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC5"> &gt; </A>]</TD>
208<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[ &lt;&lt; ]</TD>
209<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC2"> Up </A>]</TD>
210<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14"> &gt;&gt; </A>]</TD>
211<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Top">Top</A>]</TD>
212<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
213<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
214<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
215</TR></TABLE>
216<H4> 1.1.0.2 bfctsyz </H4>
217<!--docid::SEC4::-->
218Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
219<P>
220
221<DL COMPACT>
222<DT><STRONG>Usage:</STRONG>
223<DD>bfctsyz(f [,r,s,t,u,v]); f a poly, r,s,t,u optional ints, v an optional intvec
224<P>
225
226<DT><STRONG>Return:</STRONG>
227<DD>list of roots of the Bernstein-Sato polynomial bs(f) and its multiplicies
228<P>
229
230<DT><STRONG>Purpose:</STRONG>
231<DD>compute the global Bernstein-Sato polynomial for a hypersurface, defined by f, according to the algorithm by Masayuki Noro
232<P>
233
234<DT><STRONG>Note:</STRONG>
235<DD>In this proc, a system of linear equations is solved by computing syzygies.
236<BR> If r&#60;&#62;0, <CODE>std</CODE> is used for Groebner basis computations in characteristic 0,
237<BR> otherwise, and by default, <CODE>slimgb</CODE> is used.
238<BR> If s&#60;&#62;0, a matrix ordering is used for Groebner basis computations,
239<BR> otherwise, and by default, a block ordering is used.
240<BR> If t&#60;&#62;0, the minimal polynomial computation is solely performed over charasteristic 0,
241<BR> otherwise and by default, a modular method is used.
242<BR> If u&#60;&#62;0 and by default, <CODE>std</CODE> is used for Groebner basis computations in characteristic &#62;0,
243<BR> otherwise, <CODE>slimgb</CODE> is used.
244<BR> If v is a positive weight vector, v is used for homogenization computations,
245<BR> otherwise and by default, no weights are used.
246<BR> If printlevel=1, progress debug messages will be printed,
247<BR> if printlevel&#62;=2, all the debug messages will be printed.
248<P>
249
250</DL>
251<STRONG>Example:</STRONG>
252<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
253ring r = 0,(x,y),dp;
254poly f = x^2+y^3+x*y^2;
255bfctsyz(f);
256==> [1]:
257==>    _[1]=-7/6
258==>    _[2]=-1
259==>    _[3]=-5/6
260==> [2]:
261==>    1,1,1
262intvec v = 3,2;
263bfctsyz(f,0,1,1,0,v);
264==> [1]:
265==>    _[1]=-7/6
266==>    _[2]=-1
267==>    _[3]=-5/6
268==> [2]:
269==>    1,1,1
270</FONT></pre></td></tr></table><P>
271
272<A NAME="bfctonestep"></A>
273<HR SIZE="6">
274<A NAME="SEC5"></A>
275<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
276<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC4"> &lt; </A>]</TD>
277<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC6"> &gt; </A>]</TD>
278<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[ &lt;&lt; ]</TD>
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280<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14"> &gt;&gt; </A>]</TD>
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282<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
283<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
284<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
285</TR></TABLE>
286<H4> 1.1.0.3 bfctonestep </H4>
287<!--docid::SEC5::-->
288Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
289<P>
290
291<DL COMPACT>
292<DT><STRONG>Usage:</STRONG>
293<DD>bfctonestep(f [,s,t]); f a poly, s,t optional ints
294<P>
295
296<DT><STRONG>Return:</STRONG>
297<DD>list of roots of the Bernstein-Sato polynomial bs(f) and its multiplicies
298<P>
299
300<DT><STRONG>Purpose:</STRONG>
301<DD>compute the global Bernstein-Sato polynomial for a hypersurface, defined by f, using only one Groebner basis computation
302<P>
303
304<DT><STRONG>Note:</STRONG>
305<DD>If s&#60;&#62;0, <CODE>std</CODE> is used for the Groebner basis computation, otherwise,
306<BR> and by default, <CODE>slimgb</CODE> is used.
307<BR> If t&#60;&#62;0, a matrix ordering is used for Groebner basis computations,
308<BR> otherwise, and by default, a block ordering is used.
309<BR> If printlevel=1, progress debug messages will be printed,
310<BR> if printlevel&#62;=2, all the debug messages will be printed.
311<P>
312
313</DL>
314<STRONG>Example:</STRONG>
315<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
316ring r = 0,(x,y),dp;
317poly f = x^2+y^3+x*y^2;
318bfctonestep(f);
319==> [1]:
320==>    _[1]=-7/6
321==>    _[2]=-1
322==>    _[3]=-5/6
323==> [2]:
324==>    1,1,1
325bfctonestep(f,1,1);
326==> [1]:
327==>    _[1]=-7/6
328==>    _[2]=-1
329==>    _[3]=-5/6
330==> [2]:
331==>    1,1,1
332</FONT></pre></td></tr></table><P>
333
334<A NAME="bfctideal"></A>
335<HR SIZE="6">
336<A NAME="SEC6"></A>
337<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
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344<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
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346<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
347</TR></TABLE>
348<H4> 1.1.0.4 bfctideal </H4>
349<!--docid::SEC6::-->
350Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
351<P>
352
353<DL COMPACT>
354<DT><STRONG>Usage:</STRONG>
355<DD>bfctideal(I,w[,s,t]); I an ideal, w an intvec, s,t optional ints
356<P>
357
358<DT><STRONG>Return:</STRONG>
359<DD>list of roots and their multiplicies of the global b-function of I w.r.t. the weight vector (-w,w)
360<P>
361
362<DT><STRONG>Purpose:</STRONG>
363<DD>compute the global b-function of an ideal according to the algorithm by M. Noro
364<P>
365
366<DT><STRONG>Note:</STRONG>
367<DD>Assume, I is an ideal in the n-th Weyl algebra where the sequence of the
368<BR> variables is x(1),...,x(n),D(1),...,D(n).
369<BR> If s&#60;&#62;0, <CODE>std</CODE> is used for Groebner basis computations in characteristic 0,
370<BR> otherwise, and by default, <CODE>slimgb</CODE> is used.
371<BR> If t&#60;&#62;0, a matrix ordering is used for Groebner basis computations,
372<BR> otherwise, and by default, a block ordering is used.
373<BR> If printlevel=1, progress debug messages will be printed,
374<BR> if printlevel&#62;=2, all the debug messages will be printed.
375<P>
376
377</DL>
378<STRONG>Example:</STRONG>
379<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
380ring @D = 0,(x,y,Dx,Dy),dp;
381def D = Weyl();
382setring D;
383ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6;
384intvec w1 = 1,1;
385intvec w2 = 1,2;
386intvec w3 = 2,3;
387bfctideal(I,w1);
388==> [1]:
389==>    _[1]=-5/2
390==>    _[2]=-2
391==> [2]:
392==>    1,1
393bfctideal(I,w2,1);
394==> [1]:
395==>    _[1]=-10/3
396==>    _[2]=-3
397==>    _[3]=-11/3
398==> [2]:
399==>    1,1,1
400bfctideal(I,w3,0,1);
401==> [1]:
402==>    _[1]=-6
403==> [2]:
404==>    1
405</FONT></pre></td></tr></table><P>
406
407<A NAME="minpol"></A>
408<HR SIZE="6">
409<A NAME="SEC7"></A>
410<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
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417<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
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419<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
420</TR></TABLE>
421<H4> 1.1.0.5 minpol </H4>
422<!--docid::SEC7::-->
423Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
424<P>
425
426<DL COMPACT>
427<DT><STRONG>Usage:</STRONG>
428<DD>minpol(f, I); f a poly, I an ideal
429<P>
430
431<DT><STRONG>Return:</STRONG>
432<DD>coefficient vector of the minimal polynomial of the endomorphism of basering modulo I defined by f
433<P>
434
435<DT><STRONG>Purpose:</STRONG>
436<DD>compute the minimal polynomial
437<P>
438
439<DT><STRONG>Note:</STRONG>
440<DD>If f does not define an endomorphism, this proc will not terminate.
441<BR> I should be given as standard basis.
442<BR> If printlevel=1, progress debug messages will be printed,
443<BR> if printlevel&#62;=2, all the debug messages will be printed.
444<P>
445
446</DL>
447<STRONG>Example:</STRONG>
448<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
449printlevel = 0;
450ring r = 0,(x,y),dp;
451poly f = x^2+y^3+x*y^2;
452def D = initialmalgrange(f);
453setring D;
454inF;
455==> inF[1]=x*Dt
456==> inF[2]=2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy
457==> inF[3]=2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18
458==> inF[4]=18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt
459==> inF[5]=y^2*Dt
460==> inF[6]=2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6
461==> inF[7]=x*y^2+y^3+x^2
462==> inF[8]=2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+1\
463   2*y*Dy+36
464poly s = t*Dt;
465minpol(s,inF);
466==> minpol starts...
467==> with ideal I=
468==> x*Dt,
469==> 2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy,
470==> 2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18,
471==> 18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt,
472==> y^2*Dt,
473==> 2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6,
474==> x*y^2+y^3+x^2,
475==> 2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+12*y*Dy+\
476   36
477==> newNF is:
478==> t*Dt
479==> rednewNF is:
480==> t*Dt
481==> NI is:
482==> 1,
483==> t*Dt
484==> redNI is:
485==> 1,
486==> t*Dt
487==> newNF is:
488==> -1/6*y^2*Dx*Dy+1/6*y^2*Dy^2+1/36*y^2*Dx-1/36*y^2*Dy-t*Dt*Dy-1/2*x*Dx*Dy+1\
489   /6*x*Dy^2-1/3*y*Dy^2+1/12*x*Dx-1/3*y*Dx-1/36*x*Dy+13/18*y*Dy-1/18*y-4/3*D\
490   y+1/2
491==> rednewNF is:
492==> -1/6*y^2*Dx*Dy+1/6*y^2*Dy^2+1/36*y^2*Dx-1/36*y^2*Dy-t*Dt*Dy-1/2*x*Dx*Dy+1\
493   /6*x*Dy^2-1/3*y*Dy^2+1/12*x*Dx-1/3*y*Dx-1/36*x*Dy+13/18*y*Dy-1/18*y-4/3*D\
494   y+1/2
495==> NI is:
496==> 1,
497==> t*Dt,
498==> -1/6*y^2*Dx*Dy+1/6*y^2*Dy^2+1/36*y^2*Dx-1/36*y^2*Dy-t*Dt*Dy-1/2*x*Dx*Dy+1\
499   /6*x*Dy^2-1/3*y*Dy^2+1/12*x*Dx-1/3*y*Dx-1/36*x*Dy+13/18*y*Dy-1/18*y-4/3*D\
500   y+1/2
501==> redNI is:
502==> 1,
503==> t*Dt,
504==> -1/6*y^2*Dx*Dy+1/6*y^2*Dy^2+1/36*y^2*Dx-1/36*y^2*Dy-t*Dt*Dy-1/2*x*Dx*Dy+1\
505   /6*x*Dy^2-1/3*y*Dy^2+1/12*x*Dx-1/3*y*Dx-1/36*x*Dy+13/18*y*Dy-1/18*y-4/3*D\
506   y+1/2
507==> newNF is:
508==> 1/36*t*Dt
509==> rednewNF is:
510==> 0
511==> the degree of the minimal polynomial is:
512==> 3
513==> minpol finished
514==> gen(4)-1/36*gen(2)
515</FONT></pre></td></tr></table><P>
516
517<A NAME="minpolsyz"></A>
518<HR SIZE="6">
519<A NAME="SEC8"></A>
520<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
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529<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
530</TR></TABLE>
531<H4> 1.1.0.6 minpolsyz </H4>
532<!--docid::SEC8::-->
533Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
534<P>
535
536<DL COMPACT>
537<DT><STRONG>Usage:</STRONG>
538<DD>minpolsyz(f, I [,p,s,t]); f a poly, I an ideal, p, t optial ints, p a prime number
539<P>
540
541<DT><STRONG>Return:</STRONG>
542<DD>coefficient vector of the minimal polynomial of the endomorphism of basering modulo I defined by f
543<P>
544
545<DT><STRONG>Purpose:</STRONG>
546<DD>compute the minimal polynomial
547<P>
548
549<DT><STRONG>Note:</STRONG>
550<DD>If f does not define an endomorphism, this proc will not terminate.
551<BR> I should be given as standard basis.
552<BR> If p&#62;0 is given, the proc computes the minimal polynomial in char p first and
553<BR> then only searches for a minimal polynomial of the obtained degree in the basering.
554<BR> Otherwise, it searched for all degrees.
555<BR> This is done by computing syzygies.
556<BR> If s&#60;&#62;0, <CODE>std</CODE> is used for Groebner basis computations in char 0,
557<BR> otherwise, and by default, <CODE>slimgb</CODE> is used.
558<BR> If t&#60;&#62;0 and by default, <CODE>std</CODE> is used for Groebner basis computations in char &#62;0,
559<BR> otherwise, <CODE>slimgb</CODE> is used.
560<BR> If printlevel=1, progress debug messages will be printed,
561<BR> if printlevel&#62;=2, all the debug messages will be printed.
562<P>
563
564</DL>
565<STRONG>Example:</STRONG>
566<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
567printlevel = 0;
568ring r = 0,(x,y),dp;
569poly f = x^2+y^3+x*y^2;
570def D = initialmalgrange(f);
571setring D;
572inF;
573==> inF[1]=x*Dt
574==> inF[2]=2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy
575==> inF[3]=2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18
576==> inF[4]=18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt
577==> inF[5]=y^2*Dt
578==> inF[6]=2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6
579==> inF[7]=x*y^2+y^3+x^2
580==> inF[8]=2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+1\
581   2*y*Dy+36
582poly s = t*Dt;
583minpolsyz(s,inF);
584==> minpolynomial starts...
585==> with ideal I=
586==> x*Dt,
587==> 2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy,
588==> 2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18,
589==> 18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt,
590==> y^2*Dt,
591==> 2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6,
592==> x*y^2+y^3+x^2,
593==> 2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+12*y*Dy+\
594   36
595==> minpol finished
596==> gen(4)-1/36*gen(2)
597int p = prime(20000);
598minpolsyz(s,inF,p,0,0);
599==> solving in ring
600==> //   characteristic : 19997
601==> //   number of vars : 6
602==> //        block   1 : ordering dp
603==> //                  : names    t x y Dt Dx Dy
604==> //        block   2 : ordering C
605==> //   noncommutative relations:
606==> //    Dtt=t*Dt+1
607==> //    Dxx=x*Dx+1
608==> //    Dyy=y*Dy+1
609==> minpolynomial starts...
610==> with ideal I=
611==> x*Dt,
612==> 2*x*y*Dx+3*y^2*Dx-y^2*Dy-2*x*Dy,
613==> 2*x^2*Dx+x*y*Dx+x*y*Dy+18*t*Dt+9*x*Dx-x*Dy+6*y*Dy+4*x+18,
614==> 18*t*Dt^2+6*y*Dt*Dy-y*Dt+27*Dt,
615==> y^2*Dt,
616==> 2*t*y*Dt+2*x*y*Dx+2*y^2*Dx-6*t*Dt-3*x*Dx-x*Dy-2*y*Dy+2*y-6,
617==> x*y^2+y^3+x^2,
618==> 2*y^3*Dx-2*y^3*Dy-3*y^2*Dx-2*x*y*Dy+y^2*Dy-4*y^2+36*t*Dt+18*x*Dx+12*y*Dy+\
619   36
620==> minpol finished
621==> gen(4)-1/36*gen(2)
622</FONT></pre></td></tr></table><P>
623
624<A NAME="linreduce"></A>
625<HR SIZE="6">
626<A NAME="SEC9"></A>
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634<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
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636<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
637</TR></TABLE>
638<H4> 1.1.0.7 linreduce </H4>
639<!--docid::SEC9::-->
640Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
641<P>
642
643<DL COMPACT>
644<DT><STRONG>Usage:</STRONG>
645<DD>linreduce(f, I [,s]); f a poly, I an ideal, s an optional int
646<P>
647
648<DT><STRONG>Return:</STRONG>
649<DD>a poly obtained by linear reductions of the leading term of the given poly with an ideal
650<P>
651
652<DT><STRONG>Purpose:</STRONG>
653<DD>reduce a poly only by linear reductions of its leading term
654<P>
655
656<DT><STRONG>Note:</STRONG>
657<DD>If s&#60;&#62;0, a list consisting of the reduced poly and the vector of the used
658<BR> reductions is returned.
659<P>
660
661</DL>
662<STRONG>Example:</STRONG>
663<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
664ring r = 0,(x,y),dp;
665ideal I = 1,y,xy;
666poly f = 5xy+7y+3;
667poly g = 5y+7x+3;
668linreduce(f,I);
669==> 0
670linreduce(g,I);
671==> 7x+5y+3
672linreduce(f,I,1);
673==> [1]:
674==>    0
675==> [2]:
676==>    -5*gen(3)-7*gen(2)-3*gen(1)
677</FONT></pre></td></tr></table><P>
678
679<A NAME="ncsolve"></A>
680<HR SIZE="6">
681<A NAME="SEC10"></A>
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689<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
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691<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
692</TR></TABLE>
693<H4> 1.1.0.8 ncsolve </H4>
694<!--docid::SEC10::-->
695Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
696<P>
697
698<DL COMPACT>
699<DT><STRONG>Usage:</STRONG>
700<DD>ncsolve(I[,s]); I an ideal, s an optional int
701<P>
702
703<DT><STRONG>Return:</STRONG>
704<DD>coefficient vector of a linear combination of 0 in the elements of I
705<P>
706
707<DT><STRONG>Purpose:</STRONG>
708<DD>compute a linear dependency between the elements of an ideal if such one exists
709<P>
710
711<DT><STRONG>Note:</STRONG>
712<DD>If s&#60;&#62;0, <CODE>std</CODE> is used for Groebner basis computations,
713<BR> otherwise, <CODE>slimgb</CODE> is used.
714<BR> By default, <CODE>slimgb</CODE> is used in char 0 and <CODE>std</CODE> in char &#62;0.
715<BR> If printlevel=1, progress debug messages will be printed,
716<BR> if printlevel&#62;=2, all the debug messages will be printed.
717<P>
718
719</DL>
720<STRONG>Example:</STRONG>
721<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
722ring r = 0,x,dp;
723ideal I = x,2x;
724ncsolve(I);
725==> gen(2)-2*gen(1)
726ideal J = x,x2;
727ncsolve(J);
728==> 0
729</FONT></pre></td></tr></table><P>
730
731<A NAME="ispositive"></A>
732<HR SIZE="6">
733<A NAME="SEC11"></A>
734<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
735<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC10"> &lt; </A>]</TD>
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741<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
742<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
743<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
744</TR></TABLE>
745<H4> 1.1.0.9 ispositive </H4>
746<!--docid::SEC11::-->
747Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
748<P>
749
750<DL COMPACT>
751<DT><STRONG>Usage:</STRONG>
752<DD>ispositive(v); v an intvec
753<P>
754
755<DT><STRONG>Return:</STRONG>
756<DD>1 if all components of v are positive, or 0 otherwise
757<P>
758
759<DT><STRONG>Purpose:</STRONG>
760<DD>check whether all components of an intvec are positive
761<P>
762
763</DL>
764<STRONG>Example:</STRONG>
765<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
766intvec v = 1,2,3;
767ispositive(v);
768==> 1
769intvec w = 1,-2,3;
770ispositive(w);
771==> 0
772</FONT></pre></td></tr></table><P>
773
774<A NAME="isin"></A>
775<HR SIZE="6">
776<A NAME="SEC12"></A>
777<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
778<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC11"> &lt; </A>]</TD>
779<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC13"> &gt; </A>]</TD>
780<TD VALIGN="MIDDLE" ALIGN="LEFT"> &nbsp; <TD VALIGN="MIDDLE" ALIGN="LEFT">[ &lt;&lt; ]</TD>
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782<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14"> &gt;&gt; </A>]</TD>
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784<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
785<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
786<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
787</TR></TABLE>
788<H4> 1.1.0.10 isin </H4>
789<!--docid::SEC12::-->
790Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
791<P>
792
793<DL COMPACT>
794<DT><STRONG>Usage:</STRONG>
795<DD>isin(l,i); l a list, i an argument of any type
796<P>
797
798<DT><STRONG>Return:</STRONG>
799<DD>1 if i is a member of l, or 0 otherwise
800<P>
801
802<DT><STRONG>Purpose:</STRONG>
803<DD>check whether the second argument is a member of a list
804<P>
805
806</DL>
807<STRONG>Example:</STRONG>
808<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
809ring r = 0,x,dp;
810list l = 1,2,3;
811isin(l,4);
812==> 0
813isin(l,2);
814==> 1
815</FONT></pre></td></tr></table><P>
816
817<A NAME="scalarprod"></A>
818<HR SIZE="6">
819<A NAME="SEC13"></A>
820<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
821<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC12"> &lt; </A>]</TD>
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827<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
828<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
829<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
830</TR></TABLE>
831<H4> 1.1.0.11 scalarprod </H4>
832<!--docid::SEC13::-->
833Procedure from library <CODE>bfct.lib</CODE> (see  <A HREF="bfct.html#SEC2">bfct_lib</A>).
834<P>
835
836<DL COMPACT>
837<DT><STRONG>Usage:</STRONG>
838<DD>scalarprod(v,w); v,w intvecs
839<P>
840
841<DT><STRONG>Return:</STRONG>
842<DD>an int, the standard scalar product of v and w
843<P>
844
845<DT><STRONG>Purpose:</STRONG>
846<DD>compute the scalar product of two intvecs
847<P>
848
849<DT><STRONG>Note:</STRONG>
850<DD>the arguments must have the same size
851<P>
852
853</DL>
854<STRONG>Example:</STRONG>
855<TABLE><tr><td>&nbsp;</td><td class=smallexample><FONT SIZE=-1><pre>LIB "bfct.lib";
856intvec v = 1,2,3;
857intvec w = 4,5,6;
858scalarprod(v,w);
859==> 32
860</FONT></pre></td></tr></table><A NAME="Index"></A>
861<HR SIZE="6">
862<A NAME="SEC14"></A>
863<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
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870<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
871<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
872<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
873</TR></TABLE>
874<H1> 2. Index </H1>
875<!--docid::SEC14::-->
876<table><tr><th valign=top>Jump to: &nbsp; </th><td><A HREF="bfct.html#cp_B" style="text-decoration:none"><b>B</b></A>
877 &nbsp; 
878<A HREF="bfct.html#cp_I" style="text-decoration:none"><b>I</b></A>
879 &nbsp; 
880<A HREF="bfct.html#cp_L" style="text-decoration:none"><b>L</b></A>
881 &nbsp; 
882<A HREF="bfct.html#cp_M" style="text-decoration:none"><b>M</b></A>
883 &nbsp; 
884<A HREF="bfct.html#cp_N" style="text-decoration:none"><b>N</b></A>
885 &nbsp; 
886<A HREF="bfct.html#cp_S" style="text-decoration:none"><b>S</b></A>
887 &nbsp; 
888</td></tr></table><br><P></P>
889<TABLE border=0>
890<TR><TD></TD><TH ALIGN=LEFT>Index Entry</TH><TH ALIGN=LEFT> Section</TH></TR>
891<TR><TD COLSPAN=3> <HR></TD></TR>
892<TR><TH><A NAME="cp_B"></A>B</TH><TD></TD><TD></TD></TR>
893<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC3">bfct</A></TD><TD valign=top><A HREF="bfct.html#SEC3">1.1.0.1 bfct</A></TD></TR>
894<TR><TD></TD><TD valign=top><A HREF="bfct.html#IDX1">bfct.lib</A></TD><TD valign=top><A HREF="bfct.html#SEC2">1.1 bfct_lib</A></TD></TR>
895<TR><TD></TD><TD valign=top><A HREF="bfct.html#IDX2">bfct_lib</A></TD><TD valign=top><A HREF="bfct.html#SEC2">1.1 bfct_lib</A></TD></TR>
896<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC6">bfctideal</A></TD><TD valign=top><A HREF="bfct.html#SEC6">1.1.0.4 bfctideal</A></TD></TR>
897<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC5">bfctonestep</A></TD><TD valign=top><A HREF="bfct.html#SEC5">1.1.0.3 bfctonestep</A></TD></TR>
898<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC4">bfctsyz</A></TD><TD valign=top><A HREF="bfct.html#SEC4">1.1.0.2 bfctsyz</A></TD></TR>
899<TR><TD COLSPAN=3> <HR></TD></TR>
900<TR><TH><A NAME="cp_I"></A>I</TH><TD></TD><TD></TD></TR>
901<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC12">isin</A></TD><TD valign=top><A HREF="bfct.html#SEC12">1.1.0.10 isin</A></TD></TR>
902<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC11">ispositive</A></TD><TD valign=top><A HREF="bfct.html#SEC11">1.1.0.9 ispositive</A></TD></TR>
903<TR><TD COLSPAN=3> <HR></TD></TR>
904<TR><TH><A NAME="cp_L"></A>L</TH><TD></TD><TD></TD></TR>
905<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC9">linreduce</A></TD><TD valign=top><A HREF="bfct.html#SEC9">1.1.0.7 linreduce</A></TD></TR>
906<TR><TD COLSPAN=3> <HR></TD></TR>
907<TR><TH><A NAME="cp_M"></A>M</TH><TD></TD><TD></TD></TR>
908<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC7">minpol</A></TD><TD valign=top><A HREF="bfct.html#SEC7">1.1.0.5 minpol</A></TD></TR>
909<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC8">minpolsyz</A></TD><TD valign=top><A HREF="bfct.html#SEC8">1.1.0.6 minpolsyz</A></TD></TR>
910<TR><TD COLSPAN=3> <HR></TD></TR>
911<TR><TH><A NAME="cp_N"></A>N</TH><TD></TD><TD></TD></TR>
912<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC10">ncsolve</A></TD><TD valign=top><A HREF="bfct.html#SEC10">1.1.0.8 ncsolve</A></TD></TR>
913<TR><TD COLSPAN=3> <HR></TD></TR>
914<TR><TH><A NAME="cp_S"></A>S</TH><TD></TD><TD></TD></TR>
915<TR><TD></TD><TD valign=top><A HREF="bfct.html#SEC13">scalarprod</A></TD><TD valign=top><A HREF="bfct.html#SEC13">1.1.0.11 scalarprod</A></TD></TR>
916<TR><TD COLSPAN=3> <HR></TD></TR>
917</TABLE><P></P><table><tr><th valign=top>Jump to: &nbsp; </th><td><A HREF="bfct.html#cp_B" style="text-decoration:none"><b>B</b></A>
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920 &nbsp; 
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926 &nbsp; 
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928 &nbsp; 
929</td></tr></table><br><P>
930
931<HR SIZE="6">
932<A NAME="SEC_Contents"></A>
933<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
934<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Top">Top</A>]</TD>
935<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
936<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
937<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
938</TR></TABLE>
939<H1>Table of Contents</H1>
940<UL>
941<A NAME="TOC1" HREF="bfct.html#SEC1">1. Singular libraries</A>
942<BR>
943<UL>
944<A NAME="TOC2" HREF="bfct.html#SEC2">1.1 bfct_lib</A>
945<BR>
946<UL>
947<UL>
948<A NAME="TOC3" HREF="bfct.html#SEC3">1.1.0.1 bfct</A>
949<BR>
950<A NAME="TOC4" HREF="bfct.html#SEC4">1.1.0.2 bfctsyz</A>
951<BR>
952<A NAME="TOC5" HREF="bfct.html#SEC5">1.1.0.3 bfctonestep</A>
953<BR>
954<A NAME="TOC6" HREF="bfct.html#SEC6">1.1.0.4 bfctideal</A>
955<BR>
956<A NAME="TOC7" HREF="bfct.html#SEC7">1.1.0.5 minpol</A>
957<BR>
958<A NAME="TOC8" HREF="bfct.html#SEC8">1.1.0.6 minpolsyz</A>
959<BR>
960<A NAME="TOC9" HREF="bfct.html#SEC9">1.1.0.7 linreduce</A>
961<BR>
962<A NAME="TOC10" HREF="bfct.html#SEC10">1.1.0.8 ncsolve</A>
963<BR>
964<A NAME="TOC11" HREF="bfct.html#SEC11">1.1.0.9 ispositive</A>
965<BR>
966<A NAME="TOC12" HREF="bfct.html#SEC12">1.1.0.10 isin</A>
967<BR>
968<A NAME="TOC13" HREF="bfct.html#SEC13">1.1.0.11 scalarprod</A>
969<BR>
970</UL>
971</UL>
972</UL>
973<A NAME="TOC14" HREF="bfct.html#SEC14">2. Index</A>
974<BR>
975</UL>
976<HR SIZE=1>
977<A NAME="SEC_OVERVIEW"></A>
978<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
979<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Top">Top</A>]</TD>
980<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
981<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
982<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
983</TR></TABLE>
984<H1>Short Table of Contents</H1>
985<BLOCKQUOTE>
986<A NAME="TOC1" HREF="bfct.html#SEC1">1. Singular libraries</A>
987<BR>
988<A NAME="TOC14" HREF="bfct.html#SEC14">2. Index</A>
989<BR>
990
991</BLOCKQUOTE>
992<HR SIZE=1>
993<A NAME="SEC_About"></A>
994<TABLE CELLPADDING=1 CELLSPACING=1 BORDER=0>
995<TR><TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Top">Top</A>]</TD>
996<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_Contents">Contents</A>]</TD>
997<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC14">Index</A>]</TD>
998<TD VALIGN="MIDDLE" ALIGN="LEFT">[<A HREF="bfct.html#SEC_About"> ? </A>]</TD>
999</TR></TABLE>
1000<H1>About this document</H1>
1001This document was generated by <I>Oleksandr Motsak</I> on <I>November, 14  2008</I>
1002using <A HREF="http://www.mathematik.uni-kl.de/~obachman/Texi2html
1003"><I>texi2html</I></A>
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1079<TR>
1080<TD ALIGN="CENTER">
1081 [Top] </TD>
1082<TD ALIGN="CENTER">
1083Top
1084</TD>
1085<TD>
1086cover (top) of document
1087</TD>
1088<TD>
1089 &nbsp; 
1090</TD>
1091</TR>
1092<TR>
1093<TD ALIGN="CENTER">
1094 [Contents] </TD>
1095<TD ALIGN="CENTER">
1096Contents
1097</TD>
1098<TD>
1099table of contents
1100</TD>
1101<TD>
1102 &nbsp; 
1103</TD>
1104</TR>
1105<TR>
1106<TD ALIGN="CENTER">
1107 [Index] </TD>
1108<TD ALIGN="CENTER">
1109Index
1110</TD>
1111<TD>
1112concept index
1113</TD>
1114<TD>
1115 &nbsp; 
1116</TD>
1117</TR>
1118<TR>
1119<TD ALIGN="CENTER">
1120 [ ? ] </TD>
1121<TD ALIGN="CENTER">
1122About
1123</TD>
1124<TD>
1125this page
1126</TD>
1127<TD>
1128 &nbsp; 
1129</TD>
1130</TR>
1131</TABLE>
1132<P></P>
1133where the <STRONG> Example </STRONG> assumes that the current position
1134is at <STRONG> Subsubsection One-Two-Three </STRONG> of a document of
1135the following structure:
1136<UL>
1137<LI> 1. Section One  </LI>
1138<UL>
1139<LI>1.1 Subsection One-One</LI>
1140<UL>
1141<LI> ... </LI>
1142</UL>
1143<LI>1.2 Subsection One-Two</LI>
1144<UL>
1145<LI>1.2.1 Subsubsection One-Two-One
1146</LI><LI>1.2.2 Subsubsection One-Two-Two
1147</LI><LI>1.2.3 Subsubsection One-Two-Three &nbsp; &nbsp; <STRONG>
1148&lt;== Current Position </STRONG>
1149</LI><LI>1.2.4 Subsubsection One-Two-Four
1150</LI></UL>
1151<LI>1.3 Subsection One-Three</LI>
1152<UL>
1153<LI> ... </LI>
1154</UL>
1155<LI>1.4 Subsection One-Four</LI>
1156</UL>
1157</UL>
1158
1159<HR SIZE=1>
1160<BR> 
1161<FONT SIZE="-1">
1162This document was generated
1163by <I>Oleksandr Motsak</I> on <I>November, 14  2008</I>
1164using <A HREF="http://www.mathematik.uni-kl.de/~obachman/Texi2html
1165"><I>texi2html</I></A>
1166
1167</BODY>
1168</HTML>