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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 41 "Reference \+
worksheet for the gbr5 package " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
404 "The gbr5 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]);" }}}}}{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 produc
e a grlex order." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Calling Sequ
ence" }}{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 list 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 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 whose row vectors produce an elimination
 order." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Calling Sequence" }}
{PARA 0 "" 0 "" {TEXT -1 17 "elimination(k,n);" }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 10 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 40 "k = the n
umber of variables to eliminate" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 39 "n = the 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 list which represents a " }{TEXT 
271 1 "n" }{TEXT -1 3 " x " }{TEXT 272 1 "n" }{TEXT -1 113 " matrix wh
ose rows produce a matrix order which 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 " variables." }}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "el
imination(3,5);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "slowbasis_g
b()" }}{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 that indicate
s 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,...,f
s]) returns a Groebner basis of <f1,...,fs> using a naive version of B
uchberger's algorithm.  This basis is generally neither minimal nor re
duced.  Steps of constructing the Groebner basis are also printed.\n" 
}}{PARA 0 "" 0 "" {TEXT -1 121 "slowbasis_gb([f1,...,fs], nosteps) ret
urns 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]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "slowbasis_gb([x^2*y - 1, x*y
^2 - x]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "slowbasis_gb([
x^2*y - 1, x*y^2 - x], nosteps);" }}}}{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 "altb
asis_gb()" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "Purpose" }}{PARA 0 "
" 0 "" {TEXT -1 56 "altbasis_gb() finds a Groebner basis for the given
 ideal" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Calling Sequences" }}
{PARA 0 "" 0 "" {TEXT -1 25 "altbasis_gb([f1,...,fs]);" }}{PARA 0 "" 
0 "" {TEXT -1 34 "altbasis_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 -1 267 "altbasis_gb([
f1,...,fs]) returns a Groebner basis of <f1,...,fs> using a slightly m
ore insightful version of Buchberger's algorithm that slowbasis_gb(). \+
 This basis is generally neither minimal nor reduced.  Steps of constr
ucting the Groebner basis are also printed.\n" }}{PARA 0 "" 0 "" 
{TEXT -1 120 "altbasis_gb([f1,...,fs], nosteps) returns the same thing
s, 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]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "altbasis_gb([x^2*y - 1, x*y^2 - x]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "altbasis_gb([x^2*y - 1, x*y^2 - x],
 nosteps);" }}}}{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 "Purpose
" }}{PARA 0 "" 0 "" {TEXT -1 58 "quickbasis_gb() finds a Groebner basi
s for the given ideal" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Calling
 Sequences" }}{PARA 0 "" 0 "" {TEXT -1 27 "quickbasis_gb([f1,...,fs]);
" }}{PARA 0 "" 0 "" {TEXT -1 36 "quickbasis_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 define
d by ring()\n            " }}{PARA 0 "" 0 "" {TEXT -1 67 "nosteps = th
e 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> usi
ng a streamlined version of Buchberger's algorithm.  This basis is gen
erally neither minimal nor reduced.  Steps of constructing the Groebne
r basis are also printed.\n" }}{PARA 0 "" 0 "" {TEXT -1 122 "quickbasi
s_gb([f1,...,fs], nosteps) returns the same things, but steps of const
ructing the Groebner basis are not printed." }}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ri
ng(grlex, [x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "quickb
asis_gb([x^2*y - 1, x*y^2 - x]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "quickbasis_gb([x^2*y - 1, x*y^2 - x], nosteps);" }}}}
{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 b
asis." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Calling 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 ter
m ordering of ring()." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis
" }}{PARA 0 "" 0 "" {TEXT -1 111 "min_gb([g1,...,gs]) returns a list o
f polynomials that form a minimal Groebner basis for the ideal <g1,...
,gs>." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ring(lex, [x,y,z,w]);" }}}{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);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "min_gb(gb);" }}}}}{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 lis
t of polynomials that form a minimal Groebner basis under the term ord
ering of ring()." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}
{PARA 0 "" 0 "" {TEXT -1 111 "red_gb([g1,...,gs]) returns a list of po
lynomials 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]);" }}}{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));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "red_gb(gb);" }}}}{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 mult
ivariable polynomials." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Callin
g 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 polynomial to be divided" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "[f1,...,fs] = a list of p
olynomial 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 the remainder of the division with respec
t to the order and ring set by ring().  The list's second element is t
he list of quotients, with respect 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]);" }}}{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 "" }}}}{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 "quot_mx is a \+
matrix of quotients (see Synopsis)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 16 "Calling Sequence" }}{PARA 0 "" 0 "" {TEXT -1 34 "quot_mx([f1,..
.,fs], [g1,...,gt]);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Paramete
rs" }}{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 Groebner
 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 quotient
s.  In other words, we have [g1,...,gt]*Q^T = [f1,...,fs], where Q^T r
epresents the transpose of Q." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "
Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ring(grevlex, [x
,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "quot_mx([x^2*y - 1
, x*y^2 - x], [-y + x^2, y^2 - 1]);" }}}}}{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 corresponding 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 "Parameters" }}{PARA 0 "" 0 "" {TEXT -1 67 "[f1,...,fs] = \+
 a list of polynomials in the ring defined by ring()\n" }}{PARA 0 "" 
0 "" {TEXT -1 67 "nosteps = the string that indicates that no steps ar
e to be printed" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Synopsis" }}
{PARA 0 "" 0 "" {TEXT -1 545 "mxgb([f1,...,fs]) returns a list.  The f
irst 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 represent
ing the coefficient matrix, showing how each polynomial in the Groebne
r basis is represented by the polynomials <f1,...,fs>.  In other words
, [f1,...,fs]*Q^T=[g1,...,gt], where Q^T represents the transpose of Q
.  Steps of minimizing a reducing the Groebner basis and its matrix ar
e also printed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 78 "mxgb([f1,...,fs], nosteps) returns the same things, but s
teps are not printed." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ring([1,2], [x,y]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "mxgb([x^2*y - 1, x*y^2 - x])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "mxgb([x^2*y - 1, x*y^2
 - x], nosteps);" }}}{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, quickbasis_gb" }}}}}{MARK "1
3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }