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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 41 "Reference \+
worksheet for the gbm6 package " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
404 "The gbm6 package provides commands to compute Grobner bases which
 also can show the steps involved in computing them.  The major comman
ds in this package are listed below in order of need to know (i.e., th
e most basic command is first, followed by the next most basic command
, etc).  Maximize the command you would like to read about.  (To maxim
ize a command, click on the plus sign next to the command.)" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 37 "General Information about the Package" 
}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Calling Sequences" }}{PARA 0 "
" 0 "" {TEXT -1 16 "<function>(args)" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 38 "The functions in \+
the gbr5 package are:" }}{PARA 0 "" 0 "" {TEXT -1 65 " ring          l
ex\011\011      grlex\011\011       grevlex       elimination" }}
{PARA 0 "" 0 "" {TEXT -1 110 " slowbasis_gb\011        altbasis_gb\011
           quickbasis_gb\011\n\011div_alg\011\011               quot_m
x\011\011            mxgb" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 45 "This package uses the global variable morder." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Before a
ny of slowbasis_gb, altbasis_gb, quickbasis_gb, mxgb, quot_mx, or  div
_alg can be used, ring must be performed (see ring() for details)." }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "ring()" }}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 69 "ring() sets the \+
termorder and variables for the package to work under" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 17 "Calling Sequences" }}{PARA 0 "" 0 "" 
{TEXT -1 21 "ring (torder,varlist)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 68 "torder = the monomia
l order.  Valid values are lex, grlex, grevlex, " }}{PARA 0 "" 0 "" 
{TEXT -1 1 "[" }{TEXT 257 1 "k" }{TEXT -1 1 "," }{TEXT 258 1 "n" }
{TEXT -1 51 "] (the elimination order that eliminates the first " }
{TEXT 259 1 "k" }{TEXT -1 4 " of " }{TEXT 260 1 "n" }{TEXT -1 18 " var
iables), and [" }{TEXT 261 2 "v1" }{TEXT -1 4 ",..," }{TEXT 262 2 "vn
" }{TEXT -1 25 "] (a matrix order, where " }{TEXT 263 2 "vi" }{TEXT 
-1 10 " is a 1 x " }{TEXT 264 1 "n" }{TEXT -1 13 " row vector)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "varlist \+
= a list of the variables of the ring.  Note that if an elimination or
der or matrix order is used, there must be " }{TEXT 265 1 "n" }{TEXT 
-1 22 " variables in varlist." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "
Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 84 "ring(torder, varlist) returns
 the term_order with respect to the torder and varlist." }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "ring(grlex, [x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%+term_orderG" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "grlex()
" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "Creates a matrix whose row vectors produce a grlex order.
" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Calling Sequence" }}{PARA 0 
"" 0 "" {TEXT -1 8 "grlex(n)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "
Parameters" }}{PARA 0 "" 0 "" {TEXT -1 32 " n = number of variables in
 ring" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 
0 "" {TEXT -1 6 "grlex(" }{TEXT 269 1 "n" }{TEXT -1 36 ") returns a li
st which represents a " }{TEXT 266 1 "n" }{TEXT -1 3 " x " }{TEXT 267 
1 "n" }{TEXT -1 74 " matrix whose rows produce a matrix order which is
 equivalent to grlex on " }{TEXT 268 1 "n" }{TEXT -1 11 " variables." 
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "grlex(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(
7(\"\"\"F%F%F%F%F%7(F%\"\"!F'F'F'F'7(F'F%F'F'F'F'7(F'F'F%F'F'F'7(F'F'F
'F%F'F'7(F'F'F'F'F%F'" }}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 13 "elimination()" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 64 "Creates a matrix w
hose row vectors produce an elimination order." }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 17 "eli
mination(k,n);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}
{PARA 0 "" 0 "" {TEXT -1 40 "k = the number of variables to eliminate
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "n = t
he number of variables in the ring" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 12 "elimination(" }{TEXT 
274 1 "k" }{TEXT -1 1 "," }{TEXT 270 1 "n" }{TEXT -1 36 ") returns a l
ist which represents a " }{TEXT 271 1 "n" }{TEXT -1 3 " x " }{TEXT 
272 1 "n" }{TEXT -1 113 " matrix whose rows produce a matrix order whi
ch is equivalent to the elimination order that eliminates the first " 
}{TEXT 273 1 "k" }{TEXT -1 4 " of " }{TEXT 275 1 "n" }{TEXT -1 11 " va
riables." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "elimination(3,5);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#7'7'\"\"\"F%F%\"\"!F&7'F&F&F&F%F%7'F&F&F&F&!\"\"7'F&
F&F)F&F&7'F&F)F&F&F&" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "slowba
sis_gb()" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 
0 "" {TEXT -1 58 "slowbasis_gb() finds a Groebner basis for the given \+
ideal." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Calling Sequences" }}
{PARA 0 "" 0 "" {TEXT -1 26 "slowbasis_gb([f1,...,fs]);" }}{PARA 0 "" 
0 "" {TEXT -1 35 "slowbasis_gb([f1,...,fs], nosteps];" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 
79 "[f1,...,fs] =  a list of polynomials in the ring defined by ring()
\n            " }}{PARA 0 "" 0 "" {TEXT -1 67 "nosteps = the string th
at indicates that no steps are to be printed" }}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 229 "slowbasis_gb(
[f1,...,fs]) returns a Groebner basis of <f1,...,fs> using a naive ver
sion of Buchberger's algorithm.  This basis is generally neither minim
al nor reduced.  Steps of constructing the Groebner basis are also pri
nted.\n" }}{PARA 0 "" 0 "" {TEXT -1 121 "slowbasis_gb([f1,...,fs], nos
teps) returns the same things, but steps of constructing the Groebner \+
basis are not printed." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Example
s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ring(grlex, [x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_orderG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "slowbasis_gb([x^2*y - 1, x*y^2 - x]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#(%0Current~basis:~G7$,&*&)%\"xG\"\"#\"\"\"%
\"yG\"\"\"F-!\"\"F-,&*&F)F-)F,F*F+F-F)F." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#(%'Added~G7#,&%\"yG!\"\"*$)%\"xG\"\"#\"\"\"\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%3Local~divisions:~1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#(%'Added~G7$,&*$)%\"yG\"\"#\"\"\"\"\"\"!\"\"F,,&*$)F)\"
\"$F+F,F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Local~divisions:~2G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%'Added~G7\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%3Local~divisions:~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%>Total~divisions~performed:~10G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7',&*&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F+!\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,
*$F&F)F+,&*$F/F)F+F,F+,&*$)F*\"\"$F)F+F*F," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "slowbasis_gb([x^2*y - 1, x*y^2 - x], nosteps);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7',&*&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F+!
\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,*$F&F)F+,&*$F/F)F+F,F+,&*$)F*\"\"$F)F
+F*F," }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 
0 "" {TEXT -1 46 "ring(), altbasis_gb(), quickbasis_gb(), mxgb()" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "altbasis_gb()" }}{SECT 0 {PARA 4 
"" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 56 "altbasis_g
b() finds a Groebner basis for the given ideal" }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 17 "Calling Sequences" }}{PARA 0 "" 0 "" {TEXT -1 25 "al
tbasis_gb([f1,...,fs]);" }}{PARA 0 "" 0 "" {TEXT -1 34 "altbasis_gb([f
1,...,fs], nosteps];" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Paramete
rs" }}{PARA 0 "" 0 "" {TEXT -1 79 "[f1,...,fs] =  a list of polynomial
s in the ring defined by ring()\n            " }}{PARA 0 "" 0 "" 
{TEXT -1 67 "nosteps = the string that indicates that no steps are to \+
be printed" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 
0 "" 0 "" {TEXT -1 267 "altbasis_gb([f1,...,fs]) returns a Groebner ba
sis of <f1,...,fs> using a slightly more insightful version of Buchber
ger's algorithm that slowbasis_gb().  This basis is generally neither \+
minimal nor reduced.  Steps of constructing the Groebner basis are als
o printed.\n" }}{PARA 0 "" 0 "" {TEXT -1 120 "altbasis_gb([f1,...,fs],
 nosteps) returns the same things, but steps of constructing the Groeb
ner basis are not printed." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Exa
mples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ring(grlex, [x,y]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_orderG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "altbasis_gb([x^2*y - 1, x*y^2 - x]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#(%0Current~basis:~G7$,&*&)%\"xG\"\"#\"
\"\"%\"yG\"\"\"F-!\"\"F-,&*&F)F-)F,F*F+F-F)F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#(%(Added:~G7#,&%\"yG!\"\"*$)%\"xG\"\"#\"\"\"\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Local~divisions:~1G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#(%(Added:~G7#,&*$)%\"yG\"\"#\"\"\"\"\"\"!\"\"F," }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Local~divisions:~2G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#(%(Added:~G7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%3Local~divisions:~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Total~divis
ions~performed:~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*&)%\"xG\"\"
#\"\"\"%\"yG\"\"\"F+!\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,*$F&F)F+,&*$F/F)
F+F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "altbasis_gb([x^2*
y - 1, x*y^2 - x], nosteps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*&
)%\"xG\"\"#\"\"\"%\"yG\"\"\"F+!\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,*$F&F)
F+,&*$F/F)F+F,F+" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}
{PARA 0 "" 0 "" {TEXT -1 47 "ring(), slowbasis_gb(), quickbasis_gb(), \+
mxgb()" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 15 "quickbasis_gb()" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Pu
rpose" }}{PARA 0 "" 0 "" {TEXT -1 58 "quickbasis_gb() finds a Groebner
 basis for the given ideal" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Ca
lling Sequences" }}{PARA 0 "" 0 "" {TEXT -1 27 "quickbasis_gb([f1,...,
fs]);" }}{PARA 0 "" 0 "" {TEXT -1 36 "quickbasis_gb([f1,...,fs], noste
ps];" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "
" 0 "" {TEXT -1 79 "[f1,...,fs] =  a list of polynomials in the ring d
efined by ring()\n            " }}{PARA 0 "" 0 "" {TEXT -1 67 "nosteps
 = the string that indicates that no steps are to be printed" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT 
-1 236 "quickbasis_gb([f1,...,fs]) returns a Groebner basis of <f1,...
,fs> using a streamlined version of Buchberger's algorithm.  This basi
s is generally neither minimal nor reduced.  Steps of constructing the
 Groebner basis are also printed.\n" }}{PARA 0 "" 0 "" {TEXT -1 122 "q
uickbasis_gb([f1,...,fs], nosteps) returns the same things, but steps \+
of constructing the Groebner basis are not printed." }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "ring(grlex, [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_or
derG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "quickbasis_gb([x^2*
y - 1, x*y^2 - x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%0Current~Basi
s:~G7$,&*&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F-!\"\"F-,&*&F)F-)F,F*F+F-F)F." 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%(Added:~G,&%\"yG!\"\"*$)%\"xG\"\"#
\"\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%2Local~divisions:~G,
&%*localdivsG\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#(%(Added:~G
,&*$)%\"yG\"\"#\"\"\"\"\"\"!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%3Local~divisions:~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Total~divis
ions~performed:~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*&)%\"xG\"\"
#\"\"\"%\"yG\"\"\"F+!\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,*$F&F)F+,&*$F/F)
F+F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "quickbasis_gb([x^
2*y - 1, x*y^2 - x], nosteps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&
*&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F+!\"\"F+,&*&F'F+)F*F(F)F+F'F,,&F*F,*$F&
F)F+,&*$F/F)F+F,F+" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" 
}}{PARA 0 "" 0 "" {TEXT -1 46 "ring(), altbasis_gb(), quickbasis_gb(),
 mxgb()" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "min_gb()" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 29 "To
 minimize a Groebner basis." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "C
alling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 20 "min_gb([g1,...,gs]);" 
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "[g1,...,gs] = a list of polynomials that form a Groebner \+
basis under the term ordering of ring()." }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 111 "min_gb([g1,...,g
s]) returns a list of polynomials that form a minimal Groebner basis f
or the ideal <g1,...,gs>." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Exam
ples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ring(lex, [x,y,z,w])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_orderG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 82 "gb := quickbasis_gb([3*x - 6*y - 2*z, 2*x -
 4*y + 4*w, x - 2*y - z - w], nosteps);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#gbG7&,(%\"xG\"\"$%\"yG!\"'%\"zG!\"#,(F'\"\"#F)!\"%%\"wG\"\"%,
*F'\"\"\"F)F,F+!\"\"F0F4,&F+F/F0!#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "min_gb(gb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,*%
\"xG\"\"\"%\"yG!\"#%\"zG!\"\"%\"wGF*,&F)!\"%F+!#7" }}}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 8 "red_gb()" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "
Purpose" }}{PARA 0 "" 0 "" {TEXT -1 27 "To reduce a Groebner basis." }
}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 
0 "" {TEXT -1 20 "red_gb([g1,...,gs]);" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 105 "[g1,...,gs] =
 a list of polynomials that form a minimal Groebner basis under the te
rm ordering of ring()." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsi
s" }}{PARA 0 "" 0 "" {TEXT -1 111 "red_gb([g1,...,gs]) returns a list \+
of polynomials that form a minimal Groebner basis for the ideal <g1,..
.,gs>." }}}{PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "ring(lex, [x,y,z,w]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%+term_orderG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 90 "gb := min_gb(quickbasis_gb([3*x - 6*y - 2*z, 2*x - 4*y + 4*w, x \+
- 2*y - z - w], nosteps));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gbG7$
,*%\"xG\"\"\"%\"yG!\"#%\"zG!\"\"%\"wGF,,&F+!\"%F-!#7" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "red_gb(gb);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$,(%\"xG\"\"\"%\"yG!\"#%\"wG\"\"#,&%\"zGF&F)\"\"$" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 9 "div_alg()" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 72 "div_alg() performs
 the division algorithm for multivariable polynomials." }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "div_alg(f, [f1,...,fs]);" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 32 "f = the polyno
mial to be divided" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "[f1,...,fs] = a list of polynomial divisors" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" {TEXT -1 242 "
div_alg(f,[f1,...,fs]) returns a list.  The list's first element is th
e remainder of the division with respect to the order and ring set by \+
ring().  The list's second element is the list of quotients, with resp
ect to the order of [f1,...,fs]." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ring([[1,1],[
0,-1]], [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_orderG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "div_alg(5*x^2 - y, [x^2 + y,
 x^4 + 2*x^2*y + y^2 + 3]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,$%\"yG!\"'7$\"\"&\"\"!" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}{PARA 0 "" 0 "" {TEXT -1 6 "
ring()" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "quot_mx()" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "" 0 "" {TEXT -1 47 "qu
ot_mx is a matrix of quotients (see Synopsis)" }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 34 "quo
t_mx([f1,...,fs], [g1,...,gt]);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 79 "[f1,...,fs] = a list of
 polynomials, where each fi is in the ideal <g1,...,gt>\n" }}{PARA 0 "
" 0 "" {TEXT -1 120 "[g1,...,gt] = a list of polynomial that forms a G
roebner basis with respect to the monomial order and ring set by ring(
)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" 
{TEXT -1 161 "quot_mx([f1,...,fs], [g1,...,gt]) returns a matrix of qu
otients.  In other words, we have [g1,...,gt]*Q^T = [f1,...,fs], where
 Q^T represents the transpose of Q." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ring(grevl
ex, [x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+term_orderG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "quot_mx([x^2*y - 1, x*y^2 - \+
x], [-y + x^2, y^2 - 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix
G6#7$7$%\"yG\"\"\"7$\"\"!%\"xG" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
6 "mxgb()" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "
" 0 "" {TEXT -1 85 "mxgb() computes a reduced Groebner basis and its c
orresponding transformation matrix." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 17 "Calling Sequences" }}{PARA 0 "" 0 "" {TEXT -1 19 "mxgb([f1,...,
fs]); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 
"mxgb([f1,...,fs], nosteps]" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "P
arameters" }}{PARA 0 "" 0 "" {TEXT -1 67 "[f1,...,fs] =  a list of pol
ynomials in the ring defined by ring()\n" }}{PARA 0 "" 0 "" {TEXT -1 
67 "nosteps = the string that indicates that no steps are to be printe
d" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}{PARA 0 "" 0 "" 
{TEXT -1 545 "mxgb([f1,...,fs]) returns a list.  The first element, G=
[g1,...,gt], is a Groebner basis of <f1,...,fs>, using the streamlined
 version of Buchberger's algorithm of quickbasi_gb(), except that the \+
basis is reduced.  The second element a list representing the coeffici
ent matrix, showing how each polynomial in the Groebner basis is repre
sented by the polynomials <f1,...,fs>.  In other words, [f1,...,fs]*Q^
T=[g1,...,gt], where Q^T represents the transpose of Q.  Steps of mini
mizing a reducing the Groebner basis and its matrix are also printed.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "mxgb(
[f1,...,fs], nosteps) returns the same things, but steps are not print
ed." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "ring([1,2], [x,y]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%+term_orderG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "mxgb([x^2*y - 1, x*y^2 - x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%4Unminimized~basis:~G7&,&*&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F,!\"\"F,,&*&
F(F,)F+F)F*F,F(F-,&F+F-*$F'F*F,,&*$F0F*F,F-F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%5Unminimized~matrix:~G-%'matrixG6#7&7$\"\"\"\"\"!7$F*F
)7$%\"yG,$%\"xG!\"\"7$,&F)F)*$)F-\"\"#\"\"\"F0*&F-F)F/F)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%%2Minimized~basis:~G,&%\"yG!\"\"*$)%\"xG\"\"#\"
\"\"\"\"\",&*$)F%F*F+F,F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3Minim
ized~matrix:~G-%'matrixG6#7$7$%\"yG,$%\"xG!\"\"7$,&\"\"\"F/*$)F)\"\"#
\"\"\"F,*&F)F/F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,&%\"yG!\"\"
*$)%\"xG\"\"#\"\"\"\"\"\",&*$)F&F+F,F-F'F--%'matrixG6#7$7$F&,$F*F'7$,&
F-F-F/F'*&F&F-F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "mxgb(
[x^2*y - 1, x*y^2 - x], nosteps);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
$7$,&%\"yG!\"\"*$)%\"xG\"\"#\"\"\"\"\"\",&*$)F&F+F,F-F'F--%'matrixG6#7
$7$F&,$F*F'7$,&F-F-F/F'*&F&F-F*F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "See Also" }}
{PARA 0 "" 0 "" {TEXT -1 48 "ring(), slowbasis_gb, altbasis_gb, quickb
asis_gb" }}}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }