Sun Aug 18 19:02:06 2002 run on Windows
% ----------------------------------------------------------------------
% $Id: cgb.tst,v 1.2 1999/04/13 21:50:44 sturm Exp $
% ----------------------------------------------------------------------
% Copyright (c) 1999 Andreas Dolzmann and Thomas Sturm
% ----------------------------------------------------------------------
% $Log: cgb.tst,v $
% Revision 1.2 1999/04/13 21:50:44 sturm
% Removed echo, and gbverbose settings. There is no gbverbose.
%
% Revision 1.1 1999/04/05 08:30:45 dolzmann
% Copied and overworked Rev. 1.6 of the old cgb version.
%
% ----------------------------------------------------------------------
% Examples taken from the manual:
% 1 Introduction
oo := torder({x,y},lex)$
cgb {a*x,x+y};
{x + y,a*x,a*y}
gsys {a*x,x+y};
{{a <> 0,{x + y,a*x,a*y}},
{a = 0,{x + y}}}
torder oo;
{{x,y},lex}
% 4 CGB: Comprehensive Groebner Basis
oo := torder({x,y},lex)$
cgb {a*x+y,x+b*y};
{x + b*y,a*x + y,(a*b - 1)*y}
torder oo;
{{x,y},lex}
% 5 GSYS: Groebner System
oo := torder({x,y},lex)$
gsys {a*x+y,x+b*y};
{{a*b - 1 <> 0 and a <> 0,
{a*x + y,x + b*y,(a*b - 1)*y}},
{a <> 0 and a*b - 1 = 0,
{a*x + y,x + b*y}},
{a = 0,{a*x + y,x + b*y}}}
torder oo;
{{x,y},lex}
% 6 GSYS2CGB: Groebner System to CGB
oo := torder({x,y},lex)$
gsys {a*x+y,x+b*y};
{{a*b - 1 <> 0 and a <> 0,
{a*x + y,x + b*y,(a*b - 1)*y}},
{a <> 0 and a*b - 1 = 0,
{a*x + y,x + b*y}},
{a = 0,{a*x + y,x + b*y}}}
gsys2cgb ws;
{x + b*y,a*x + y,(a*b - 1)*y}
torder oo;
{{x,y},lex}
% 7 Switch CGBREAL: Computing over the Real Numbers
oo := torder({x,y},lex)$
off cgbreal;
gsys {a*x+y,x-a*y};
2
{{a + 1 <> 0 and a <> 0,
2
{a*x + y,x - a*y,(a + 1)*y}},
2
{a <> 0 and a + 1 = 0,{a*x + y,x - a*y}},
{a = 0,{a*x + y,x - a*y}}}
on cgbreal;
gsys({a*x+y,x-a*y});
{{a <> 0,
2
{a*x + y,x - a*y,(a + 1)*y}},
{a = 0,{a*x + y,x - a*y}}}
torder oo;
{{x,y},lex}
% Miscellaneous examples:
% Dolzmann's Example
oo := torder({x,y,z},lex);
oo := {{},lex}
cgb({a*x+b*y,c*x+d*y,(a*d-b*c)*z});
{c*x + d*y,
a*x + b*y,
(a*d - b*c)*y,
(a*d - b*c)*z}
gsys({a*x+b*y,c*x+d*y,(a*d-b*c)*z});
{{a*d - b*c <> 0 and a <> 0 and c <> 0,
{a*x + b*y,
c*x + d*y,
(a*d - b*c)*y,
(a*d - b*c)*z}},
{a*d - b*c <> 0 and a <> 0 and d <> 0 and c = 0,
{a*x + b*y,c*x + d*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and a <> 0 and c = 0 and d = 0,
{a*x + b*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and b <> 0 and c <> 0 and a = 0,
{a*x + b*y,c*x + d*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and b <> 0 and d <> 0 and a = 0 and c = 0,
{a*x + b*y,c*x + d*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and b <> 0 and a = 0 and c = 0 and d = 0,
{a*x + b*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and c <> 0 and a = 0 and b = 0,
{c*x + d*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and d <> 0 and a = 0 and b = 0 and c = 0,
{c*x + d*y,(a*d - b*c)*z}},
{a*d - b*c <> 0 and a = 0 and b = 0 and c = 0 and d = 0,
{(a*d - b*c)*z}},
{a <> 0 and c <> 0 and a*d - b*c = 0,
{a*x + b*y,c*x + d*y}},
{a <> 0 and d <> 0 and a*d - b*c = 0 and c = 0,
{a*x + b*y,c*x + d*y}},
{a <> 0 and a*d - b*c = 0 and c = 0 and d = 0,
{a*x + b*y}},
{b <> 0 and c <> 0 and a*d - b*c = 0 and a = 0,
{a*x + b*y,c*x + d*y}},
{b <> 0 and d <> 0 and a*d - b*c = 0 and a = 0 and c = 0,
{a*x + b*y,c*x + d*y}},
{b <> 0 and a*d - b*c = 0 and a = 0 and c = 0 and d = 0,
{a*x + b*y}},
{c <> 0 and a*d - b*c = 0 and a = 0 and b = 0,
{c*x + d*y}},
{d <> 0 and a*d - b*c = 0 and a = 0 and b = 0 and c = 0,
{c*x + d*y}},
{a*d - b*c = 0 and a = 0 and b = 0 and c = 0 and d = 0,
{}}}
gsys2cgb ws;
{c*x + d*y,
a*x + b*y,
(a*d - b*c)*y,
(a*d - b*c)*z}
torder oo;
{{x,y,z},lex}
% Forsman's Example (hybrid control system).
oo := torder({x1,x2,y2,y1,y0},lex);
oo := {{},lex}
gsys({(u1*u2-u1)*x1+u2*x2+y2,(u2-1)*x1+u2*x2+y1,-x2+y0});
{{u1*u2 - u1 <> 0,
{(u1*u2 - u1)*x1 + u2*x2 + y2,
(u2 - 1)*x1 + u2*x2 + y1,
x2 - y0,
y2 - u1*y1 - (u1*u2 - u2)*y0}},
{u2 - 1 <> 0 and u2 <> 0 and u1*u2 - u1 = 0,
{(u1*u2 - u1)*x1 + u2*x2 + y2,
(u2 - 1)*x1 + u2*x2 + y1,
(u1*u2 - u1)*x1 + y2 + u2*y0,
x2 - y0}},
{u1*u2 - u1 = 0 and u2 - 1 = 0,
{(u1*u2 - u1)*x1 + u2*x2 + y2,
(u2 - 1)*x1 + u2*x2 + y1,
(u1*u2 - u1 - u2 + 1)*x1 + y2 - y1,
(u2 - 1)*x1 + y1 + u2*y0,
x2 - y0}},
{u1*u2 - u1 = 0 and u2 = 0,
{(u1*u2 - u1)*x1 + u2*x2 + y2,
(u2 - 1)*x1 + u2*x2 + y1,
x2 - y0}}}
torder oo;
{{x1,x2,y2,y1,y0},lex}
% Weispfenning's Example
oo := torder({x,y},lex);
oo := {{},lex}
gsys({v*x*y + x,u*y^2 + x^2});
{{u*v <> 0 and v <> 0,
2 2
{x + u*y ,
v*x*y + x,
3 2
(u*v)*y + u*y }},
{u <> 0 and v <> 0 and u*v = 0,
2 2
{x + u*y ,
3 3
(u*v )*x*y + u*x,
v*x*y + x,
3 2
(u*v)*y + u*y }},
{u <> 0 and v = 0,
2 2
{v*x *y - u*y ,
2 2
x + u*y ,
v*x*y + x}},
{v <> 0 and u*v = 0 and u = 0,
2 2
{x + u*y ,v*x*y + x}},
{u = 0 and v = 0,
2 2
{x + u*y ,v*x*y + x}}}
torder oo;
{{x,y},lex}
% The folllowing three examples are taken from
% Weispfenning, Comprehensive Groebner Bases,
% J. Symbolic Computation (1992) 14, 1-29
% Weispfenning's Example 7.1
oo := torder({x},lex);
oo := {{},lex}
gsys({a0*x**2 + a1*x + a2,b0*x**2 + b1*x + b2});
3 2 2 2 2 2 2
{{a0 *b2 - a0 *a1*b1*b2 - 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2
2 2
- a0*a1*a2*b0*b1 + a0*a2 *b0 <> 0 and a0*b1 - a1*b0 <> 0 and a0 <> 0
and b0 <> 0,
3 2 2 2 2 2 2
{a0 *b2 - a0 *a1*b1*b2 - 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2
2 2
- a0*a1*a2*b0*b1 + a0*a2 *b0 }},
3 2 2
{a0*b1 - a1*b0 <> 0 and a0 <> 0 and b0 <> 0 and a0 *b2 - a0 *a1*b1*b2
2 2 2 2 2 2
- 2*a0 *a2*b0*b2 + a0 *a2*b1 + a0*a1 *b0*b2 - a0*a1*a2*b0*b1 + a0*a2 *b0
= 0,
2
{a0*x + a1*x + a2,
2
b0*x + b1*x + b2,
(a0*b1 - a1*b0)*x + (a0*b2 - a2*b0)}},
2 2
{a0*b2 - a1*b1*b2 + a2*b1 <> 0 and a0 <> 0 and b1 <> 0 and b0 = 0,
3 2 2 2
{(a0*b0*b1)*x - (a0*b0*b2 - a1*b0*b1)*x - (a0*b2 - a1*b1*b2 + a2*b1 )}},
{a0*b2 - a2*b0 <> 0 and a0 <> 0 and b0 <> 0 and a0*b1 - a1*b0 = 0,
{(a0*b1 - a1*b0)*x + (a0*b2 - a2*b0)}},
{a0 <> 0 and b0 <> 0 and a0*b1 - a1*b0 = 0 and a0*b2 - a2*b0 = 0,
2 2
{a0*x + a1*x + a2,b0*x + b1*x + b2}},
2 2
{a0 <> 0 and b1 <> 0 and a0*b2 - a1*b1*b2 + a2*b1 = 0 and b0 = 0,
2 2
{a0*x + a1*x + a2,b0*x + b1*x + b2}},
{a0 <> 0 and b2 <> 0 and b0 = 0 and b1 = 0,
2
{b0*x + b1*x + b2}},
{a0 <> 0 and b0 = 0 and b1 = 0 and b2 = 0,
2
{a0*x + a1*x + a2}},
2 2
{a1 *b2 - a1*a2*b1 + a2 *b0 <> 0 and a1 <> 0 and b0 <> 0 and a0 = 0,
3 2 2 2
{(a0*a1*b0)*x + (a0*a1*b1 - a0*a2*b0)*x - (a1 *b2 - a1*a2*b1 + a2 *b0)}},
{a1*b2 - a2*b1 <> 0 and a1 <> 0 and b1 <> 0 and a0 = 0 and b0 = 0,
2
{(a0*b1 - a1*b0)*x - (a1*b2 - a2*b1)}},
2 2
{a1 <> 0 and b0 <> 0 and a0 = 0 and a1 *b2 - a1*a2*b1 + a2 *b0 = 0,
2 2
{a0*x + a1*x + a2,b0*x + b1*x + b2}},
{a1 <> 0 and b1 <> 0 and a0 = 0 and a1*b2 - a2*b1 = 0 and b0 = 0,
2 2
{a0*x + a1*x + a2,b0*x + b1*x + b2}},
{a1 <> 0 and b2 <> 0 and a0 = 0 and b0 = 0 and b1 = 0,
2
{b0*x + b1*x + b2}},
{a1 <> 0 and a0 = 0 and b0 = 0 and b1 = 0 and b2 = 0,
2
{a0*x + a1*x + a2}},
{a2 <> 0 and a0 = 0 and a1 = 0,
2
{a0*x + a1*x + a2}},
{b0 <> 0 and a0 = 0 and a1 = 0 and a2 = 0,
2
{b0*x + b1*x + b2}},
{b1 <> 0 and a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0,
2
{b0*x + b1*x + b2}},
{b2 <> 0 and a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0 and b1 = 0,
2
{b0*x + b1*x + b2}},
{a0 = 0 and a1 = 0 and a2 = 0 and b0 = 0 and b1 = 0 and b2 = 0,
{}}}
torder oo;
{{x},lex}
% Weispfenning's Example 7.2
oo := torder({x,y},lex);
oo := {{},lex}
gsys({v*x*y + u*x**2 + x,u*y**2 + x**2});
4 3
{{u *v + u*v <> 0 and u <> 0 and v <> 0,
2
{u*x + v*x*y + x,
2 2
x + u*y ,
2 2
v*x*y + x - u *y ,
3 5 2 3 3 5 2 2 2
u *x + (u *v + u *v )*y - (u - u *v )*y ,
7 2 4 4 4 4 3 3 4 2 2
(u *v + u *v )*y + (2*u *v )*y + (u *v )*y }},
4 3
{u <> 0 and v <> 0 and u *v + u*v = 0,
2
{u*x + v*x*y + x,
2 2
x + u*y ,
2 2
v*x*y + x - u *y ,
3 5 2 3 3 5 2 2 2
u *x + (u *v + u *v )*y - (u - u *v )*y ,
7 2 4 4 4 4 3 3 4 2 2
(u *v + u *v )*y + (2*u *v )*y + (u *v )*y }},
{u <> 0 and v = 0,
2 3 3 5 4 2 2
{(u*v)*x *y + (u *v)*x*y - u *y - u *y ,
2
u*x + v*x*y + x,
2 2
x + u*y ,
2 2
v*x*y + x - u *y }},
{v <> 0 and u = 0,
2 2 2
{u*x + v*x*y + x,x + u*y }},
{u = 0 and v = 0,
2 2 2
{u*x + v*x*y + x,x + u*y }}}
torder oo;
{{x,y},lex}
% Weispfenning's Example 7.3
oo := torder({x1,x2,x3,x4},lex);
oo := {{},lex}
gsys {x4 - (a4-a2),x1 + x2 + x3 + x4 + (a1 + a3 + a4),
x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a4 + a1*a3 + a3*a4),x1*x3*x4 - a1*a3*a4};
2 2 3 2 2
{{a1*a2 - a1*a2*a3 - 3*a1*a2*a4 + 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4
2 2 3
- 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4 + 2*a4 <> 0 and a2 - a4 = 0,
2
{(a2 - a4)*x2*x3 - (a2 - a4)*x2*x4 + (a1*a2 - a1*a2*a3 - 3*a1*a2*a4
2 3 2 2 2 2
+ 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4 - 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4
3
+ 2*a4 )}},
{a2 - a4 <> 0,
{x1*x3*x4 - a1*a3*a4,
x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a3 + a1*a4 + a3*a4),
x1 + x2 + x3 + x4 + (a1 + a3 + a4),
2
(a2 - a4)*x2 - x3 - (a1 - a2 + a3 + 2*a4)*x3
2 2
+ (a1*a2 - a1*a3 - 2*a1*a4 - a2 + a2*a3 + 3*a2*a4 - 2*a3*a4 - 2*a4 ),
3 2
x3 + (a1 + a3 + a4)*x3 + (a1*a3 + a1*a4 + a3*a4)*x3 - a1*a3*a4,
x4 + (a2 - a4)}},
2 2 3 2 2
{a1*a2 - a1*a2*a3 - 3*a1*a2*a4 + 2*a1*a4 - a2 + a2 *a3 + 4*a2 *a4
2 2 3
- 3*a2*a3*a4 - 5*a2*a4 + 2*a3*a4 + 2*a4 = 0 and a2 - a4 = 0,
{x1*x3*x4 - a1*a3*a4,
x1*x3 + x1*x4 + x2*x3 + x3*x4 - (a1*a3 + a1*a4 + a3*a4),
x1 + x2 + x3 + x4 + (a1 + a3 + a4),
2
(a2 - a4)*x2 - x3 - (a1 - a2 + a3 + 2*a4)*x3
2 2
+ (a1*a2 - a1*a3 - 2*a1*a4 - a2 + a2*a3 + 3*a2*a4 - 2*a3*a4 - 2*a4 ),
x4 + (a2 - a4)}}}
torder oo;
{{x1,x2,x3,x4},lex}
% Pesch's example (Circle through three points)
oo := torder({y,x},revgradlex);
oo := {{},lex}
gsys({2*b2*y + 2*a2*x - b2**2 + a2**2,2*b1*y + 2*a1*x - b1**2 + a1**2});
2 2 2 2
{{a1 *a2 - a1*a2 + a1*b2 - a2*b1 <> 0 and a1 <> 0 and a2 <> 0 and b1 = 0
and b2 = 0,
2 2 2 2
{(2*a1*b2 - 2*a2*b1)*y - (a1 *a2 - a1*a2 + a1*b2 - a2*b1 )}},
2 2 2 2
{a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 <> 0 and b1 <> 0 and b2 <> 0
and a1*b2 - a2*b1 = 0,
2 2 2 2
{(2*a1*b2 - 2*a2*b1)*x + (a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 )}},
2 2
{a1 - b1 <> 0 and a1 = 0 and b1 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 )}},
{a1*b2 - a2*b1 <> 0 and b1 <> 0 and b2 <> 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 ),
2 2
(2*b2)*y + (2*a2)*x + (a2 - b2 ),
2 2 2 2
(2*a1*b2 - 2*a2*b1)*x + (a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 )}},
2 2
{a1 <> 0 and a2 - b2 <> 0 and a2 = 0 and b1 = 0 and b2 = 0,
2 2
{(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2 2 2
{a1 <> 0 and a2 <> 0 and a1 *a2 - a1*a2 + a1*b2 - a2*b1 = 0 and b1 = 0
and b2 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 ),
2 2
(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
{a1 <> 0 and b2 <> 0 and b1 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 ),
2 2
(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2
{a1 <> 0 and a2 - b2 = 0 and a2 = 0 and b1 = 0 and b2 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 )}},
2 2
{a2 - b2 <> 0 and b1 <> 0 and a2 = 0 and b2 = 0,
2 2
{(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2 2 2
{a2 - b2 <> 0 and a1 - b1 = 0 and a1 = 0 and a2 = 0 and b1 = 0 and b2 = 0,
2 2
{(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
{a2 <> 0 and b1 <> 0 and b2 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 ),
2 2
(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2
{a2 <> 0 and a1 - b1 = 0 and a1 = 0 and b1 = 0 and b2 = 0,
2 2
{(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2 2 2
{b1 <> 0 and b2 <> 0 and a1 *b2 - a2 *b1 - b1 *b2 + b1*b2 = 0
and a1*b2 - a2*b1 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 ),
2 2
(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2
{b1 <> 0 and a2 - b2 = 0 and a2 = 0 and b2 = 0,
2 2
{(2*b1)*y + (2*a1)*x + (a1 - b1 )}},
2 2
{b2 <> 0 and a1 - b1 = 0 and a1 = 0 and b1 = 0,
2 2
{(2*b2)*y + (2*a2)*x + (a2 - b2 )}},
2 2 2 2
{a1 - b1 = 0 and a1 = 0 and a2 - b2 = 0 and a2 = 0 and b1 = 0 and b2 = 0,
{}}}
torder oo;
{{y,x},revgradlex}
% Effelterre's example (Aspect graphs)
f1 := -4-4*v**2-4*u**2+40*v*v1+24*v-120*v1+8*u-40*v2-68*v1**2-100*v2**2+40*u*v2+
24*v1*v2-4*v1**2*u-4*v2**2*v**2+24*v2**2*v-24*v1*u*v2+8*v*v1*u*v2$
f2 := 8*v*v1*u*v2-4*v1**2*u**2+4*v1**2-4*v2**2*v**2+4*v2**2-16*v**2-16*u**2+16$
f3 := 16*v-48*u+16*v*v1**2-48*u*v2**2-12*v1**2*u+4*v2**2*v-36*v*v1*v2+
12*v1*u*v2+12*v*v2**2*u-
80*u*v1+80*v2*v-20*v1*u*v2**2+20*v2*v*v1**2-20*v1**3*u+20*v2**3*v-12*v1**2*v*u+
12*v2*v**2*v1-12*v1*u**2*v2$
f4 := -160u*v2-1596v2**2+3200*v2-1596-4*u**2+160*u$
% Special case I2, v1=0
oo := torder({v,u},lex);
oo := {{},lex}
gsys(sub(v1=0,{f1,f2,f4}));
16 15 14
{{1000557799569*v2 - 8971728968760*v2 + 32332553961916*v2
13 12 11
- 56816638983720*v2 + 43557281160966*v2 - 2232864400680*v2
10 9 8
- 2141149653636*v2 - 24710286679320*v2 + 22557177598385*v2
7 6 5
- 4364899181280*v2 + 1970637124608*v2 - 2915313822720*v2
4 3 2
+ 1467702460416*v2 - 1340557025280*v2 + 685570158592*v2 - 140796887040*v2
7 6 5 4
+ 63647907840 <> 0 and 55175*v2 - 184787*v2 + 144885*v2 + 44895*v2
3 2
- 13020*v2 - 35580*v2 + 320*v2 - 6368 <> 0,
16 15 14
{1000557799569*v2 - 8971728968760*v2 + 32332553961916*v2
13 12 11
- 56816638983720*v2 + 43557281160966*v2 - 2232864400680*v2
10 9 8
- 2141149653636*v2 - 24710286679320*v2 + 22557177598385*v2
7 6 5
- 4364899181280*v2 + 1970637124608*v2 - 2915313822720*v2
4 3 2
+ 1467702460416*v2 - 1340557025280*v2 + 685570158592*v2 - 140796887040*v2
+ 63647907840}},
8 7 6 5 4
{3389663*v2 - 14658420*v2 + 20230730*v2 - 6945060*v2 - 3218385*v2
3 2 7
- 497040*v2 + 1656704*v2 - 257280*v2 + 255872 <> 0 and 55175*v2
6 5 4 3 2
- 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2 + 320*v2 - 6368
= 0,
7 6 5 4 3 2
{(220700*v2 - 739148*v2 + 579540*v2 + 179580*v2 - 52080*v2 - 142320*v2
8 7 6
+ 1280*v2 - 25472)*u + (3389663*v2 - 14658420*v2 + 20230730*v2
5 4 3 2
- 6945060*v2 - 3218385*v2 - 497040*v2 + 1656704*v2 - 257280*v2
+ 255872)}},
7 6 5 4 3 2
{55175*v2 - 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2
16 15
+ 320*v2 - 6368 <> 0 and 1000557799569*v2 - 8971728968760*v2
14 13 12
+ 32332553961916*v2 - 56816638983720*v2 + 43557281160966*v2
11 10 9
- 2232864400680*v2 - 2141149653636*v2 - 24710286679320*v2
8 7 6
+ 22557177598385*v2 - 4364899181280*v2 + 1970637124608*v2
5 4 3
- 2915313822720*v2 + 1467702460416*v2 - 1340557025280*v2
2
+ 685570158592*v2 - 140796887040*v2 + 63647907840 = 0,
2 2 2 2 2
{(v2 + 1)*v - (6*v2 + 6)*v + u - (10*v2 + 2)*u + (25*v2 + 10*v2 + 1),
2 2 2 2
(v2 + 4)*v + 4*u - (v2 + 4),
4 2 3 2
(6*v2 + 30*v2 + 24)*v - (110*v2 - 122*v2 - 40*v2 - 8)*u
4 3 2
- (1223*v2 - 2390*v2 + 1303*v2 + 40*v2 + 8),
2 2
u + (40*v2 - 40)*u + (399*v2 - 800*v2 + 399),
7 6 5 4 3 2
(220700*v2 - 739148*v2 + 579540*v2 + 179580*v2 - 52080*v2 - 142320*v2
8 7 6
+ 1280*v2 - 25472)*u + (3389663*v2 - 14658420*v2 + 20230730*v2
5 4 3 2
- 6945060*v2 - 3218385*v2 - 497040*v2 + 1656704*v2 - 257280*v2
+ 255872)}},
8 7 6 5 4
{3389663*v2 - 14658420*v2 + 20230730*v2 - 6945060*v2 - 3218385*v2
3 2 7
- 497040*v2 + 1656704*v2 - 257280*v2 + 255872 = 0 and 55175*v2
6 5 4 3 2
- 184787*v2 + 144885*v2 + 44895*v2 - 13020*v2 - 35580*v2 + 320*v2 - 6368
= 0,
2 2 2 2 2
{(v2 + 1)*v - (6*v2 + 6)*v + u - (10*v2 + 2)*u + (25*v2 + 10*v2 + 1),
2 2 2 2
(v2 + 4)*v + 4*u - (v2 + 4),
4 2 3 2
(6*v2 + 30*v2 + 24)*v - (110*v2 - 122*v2 - 40*v2 - 8)*u
4 3 2
- (1223*v2 - 2390*v2 + 1303*v2 + 40*v2 + 8),
2 2
u + (40*v2 - 40)*u + (399*v2 - 800*v2 + 399)}}}
torder oo;
{{v,u},lex}
clear f1,f2,f3,f4;
% Sit's Example 2.2
oo := torder({z2,z2},revgradlex);
oo := {{},lex}
gsys({d*z2 + c*z1 - v,b*z2 + a*z1 - u});
{{a*d*z1 - b*c*z1 + b*v - d*u <> 0 and b <> 0 and d <> 0,
{a*d*z1 - b*c*z1 + b*v - d*u}},
{a*z1 - u <> 0 and b = 0,
{b*z2 + (a*z1 - u)}},
{b <> 0 and c*z1 - v <> 0 and d = 0,
{d*z2 + (c*z1 - v)}},
{b <> 0 and d <> 0 and a*d*z1 - b*c*z1 + b*v - d*u = 0,
{b*z2 + (a*z1 - u),d*z2 + (c*z1 - v)}},
{b <> 0 and c*z1 - v = 0 and d = 0,
{b*z2 + (a*z1 - u)}},
{c*z1 - v <> 0 and a*z1 - u = 0 and b = 0 and d = 0,
{d*z2 + (c*z1 - v)}},
{d <> 0 and a*z1 - u = 0 and b = 0,
{d*z2 + (c*z1 - v)}},
{a*z1 - u = 0 and b = 0 and c*z1 - v = 0 and d = 0,
{}}}
torder oo;
{{z2,z2},revgradlex}
% Sit's Example 2.3
oo := torder({z2,z2},revgradlex);
oo := {{},lex}
gsys({x**3*z2 + (x**2+1)*z1,x**2*z2 + x*z1 - 1});
2
{{x *z1 + z1 <> 0 and x = 0,
3 2
{x *z2 + (x *z1 + z1)}},
2
{x*z1 - 1 <> 0 and x *z1 + z1 = 0 and x = 0,
2
{x *z2 + (x*z1 - 1)}},
{x + z1 <> 0 and x <> 0,{x + z1}},
{x <> 0 and x + z1 = 0,
3 2 2
{x *z2 + (x *z1 + z1),x *z2 + (x*z1 - 1)}},
2
{x *z1 + z1 = 0 and x*z1 - 1 = 0 and x = 0,{}}}
torder oo;
{{z2,z2},revgradlex}
% Sit's Example 3.3
oo := torder({z3,z2,z2},revgradlex);
oo := {{},lex}
gsys({z3 + b*z2 + a*z1 - 1,a*z3 + z2 + b*z1 - 1,b*z3 + a*z2 + z1 - 1});
3 2
{{a *z1 - a - 2*a*b*z1 + a*b + a + z1 - 1 <> 0 and a <> 0 and b = 0,
2 3 2
{(a*b - b)*z3 - (a *z1 - a - 2*a*b*z1 + a*b + a + z1 - 1)}},
3 2 3 2 2
{a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1 <> 0 and a - b <> 0
and a <> 0 and b <> 0,
3 2 3 2
{a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1}},
2
{a *z1 - a - b*z1 + 1 <> 0 and a <> 0 and b <> 0 and a*b*z1 - b - z1 + 1 = 0
2
and a*b - 1 = 0 and a - b = 0,
2
{(a*b - 1)*z2 + (a *z1 - a - b*z1 + 1)}},
3 2
{2*a*b*z1 - a - b *z1 + b - b - z1 + 1 <> 0 and b <> 0 and a = 0,
2 2 3 2
{(a - a*b )*z3 + (2*a*b*z1 - a - b *z1 + b - b - z1 + 1)}},
2
{a*b*z1 - b - z1 + 1 <> 0 and a <> 0 and b <> 0 and a - b = 0,
2
{(a - b )*z2 - (a*b*z1 - b - z1 + 1)}},
2
{a*b - 1 <> 0 and a <> 0 and b <> 0 and a*b*z1 - b - z1 + 1 = 0 and a - b = 0,
{z3 + b*z2 + (a*z1 - 1),
b*z3 + a*z2 + (z1 - 1),
a*z3 + z2 + (b*z1 - 1),
2
(a*b - 1)*z2 + (a *z1 - a - b*z1 + 1)}},
2
{a - b <> 0 and a <> 0 and b <> 0
3 2 3 2
and a *z1 - a - 3*a*b*z1 + a*b + a + b *z1 - b + b + z1 - 1 = 0,
{z3 + b*z2 + (a*z1 - 1),
b*z3 + a*z2 + (z1 - 1),
a*z3 + z2 + (b*z1 - 1),
2
(a - b )*z2 - (a*b*z1 - b - z1 + 1)}},
2
{a <> 0 and b <> 0 and a *z1 - a - b*z1 + 1 = 0 and a*b*z1 - b - z1 + 1 = 0
2
and a*b - 1 = 0 and a - b = 0,
{z3 + b*z2 + (a*z1 - 1),
b*z3 + a*z2 + (z1 - 1),
a*z3 + z2 + (b*z1 - 1)}},
3 2
{a <> 0 and a *z1 - a - 2*a*b*z1 + a*b + a + z1 - 1 = 0 and b = 0,
{z3 + b*z2 + (a*z1 - 1),
b*z3 + a*z2 + (z1 - 1),
a*z3 + z2 + (b*z1 - 1)}},
3 2
{b <> 0 and 2*a*b*z1 - a - b *z1 + b - b - z1 + 1 = 0 and a = 0,
{z3 + b*z2 + (a*z1 - 1),
b*z3 + a*z2 + (z1 - 1),
a*z3 + z2 + (b*z1 - 1)}},
{z1 - 1 <> 0 and a = 0 and b = 0,
{b*z3 + a*z2 + (z1 - 1)}},
{a = 0 and b = 0 and z1 - 1 = 0,
{z3 + b*z2 + (a*z1 - 1),a*z3 + z2 + (b*z1 - 1)}}}
torder oo;
{{z3,z2,z2},revgradlex}
% Sit's Example 8.3
oo := torder({z4,z3,z2,z2},revgradlex);
oo := {{},lex}
gsys({z4 + c*z3 + b*z2 + a*z1 - w2,2*z4 + z2 - w1,a*z4 - z3 - w4,d*z4 + z3 +
2*z1 - w3,z4 + z1 - w5});
3 2 2 2 2 2 2 2 2
{{a *c*z1 + a *b*c*w1 + a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1
2
+ a*b*c*d*w1 - 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4
- a*c*d*w2 + 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3
- 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 <> 0 and a*b + b*d <> 0
and a*c + 1 <> 0 and a <> 0 and d <> 0,
3 2 2 2 2 2 2 2 2
{a *c*z1 + a *b*c*w1 + a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1
2
+ a*b*c*d*w1 - 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4
- a*c*d*w2 + 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3
- 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1}},
3 2 2 2 2
{a *c*z1 + a *b*c*w1 - a *c*w2 - a *c*w5 + a *c*z1 - 2*a*b*c*w4 + 2*a*b*w5
- 2*a*b*z1 + 2*a*c*w4 - a*w5 + a*z1 - 2*b*w4 + w4 <> 0 and a*b <> 0
and a*c - 2*b + 1 <> 0 and a <> 0 and d <> 0 and a*c + 1 = 0
and a*w3 - 2*a*z1 - d*w4 = 0 and a + d = 0,
2 2 3 2 2
{(a *c - 2*a*b*c + 2*a*c - 2*b + 1)*z3 + (a *c*z1 + a *b*c*w1 - a *c*w2
2 2
- a *c*w5 + a *c*z1 - 2*a*b*c*w4 + 2*a*b*w5 - 2*a*b*z1 + 2*a*c*w4 - a*w5
+ a*z1 - 2*b*w4 + w4)}},
3 2 2 2 2 2
{a *z1 + 2*a *b*c*w3 - 4*a *b*c*z1 + a *b*w1 + a *d*z1 - a *w2 - 2*a*b*c*d*w4
+ a*b*d*w1 - 2*a*b*w4 - a*d*w2 + a*w4 - 2*b*d*w4 + d*w4 <> 0 and a*b <> 0
and a + d <> 0 and a <> 0 and d <> 0 and a*c + 1 = 0,
2 2 3
{(2*a *b*c - a *c + 2*a*b*c*d + 2*a*b - a*c*d - a + 2*b*d - d)*z3 - (a *z1
2 2 2 2 2
+ 2*a *b*c*w3 - 4*a *b*c*z1 + a *b*w1 + a *d*z1 - a *w2 - 2*a*b*c*d*w4
+ a*b*d*w1 - 2*a*b*w4 - a*d*w2 + a*w4 - 2*b*d*w4 + d*w4)}},
2 2 2 2 2 2
{a *c *w5 - a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4
- a*c*w2 + 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2
+ w5 - z1 <> 0 and a*c - 2*b + 1 <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0
2
and a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1
= 0 and a*b + b*d = 0,
2 2 2 2 2 2
{a *c *w5 - a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4
- a*c*w2 + 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2
+ w5 - z1}},
2 2 2 2
{a *c*w3 - 2*a *c*z1 + a *w5 - a *z1 - a*c*d*w4 + a*d*w5 - a*d*z1 - a*w4 - d*w4
3
<> 0 and a*b <> 0 and a + d <> 0 and a <> 0 and d <> 0 and a *z1
2 2 2 2 2
+ 2*a *b*c*w3 - 4*a *b*c*z1 + a *b*w1 + a *d*z1 - a *w2 - 2*a*b*c*d*w4
+ a*b*d*w1 - 2*a*b*w4 - a*d*w2 + a*w4 - 2*b*d*w4 + d*w4 = 0 and a*c + 1 = 0,
2 2 2 2 2
{(a *c + a*c*d + a + d)*z3 - (a *c*w3 - 2*a *c*z1 + a *w5 - a *z1 - a*c*d*w4
+ a*d*w5 - a*d*z1 - a*w4 - d*w4)}},
2 2
{a *c*w5 - a *c*z1 + a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5
- a*z1 + d*w5 - d*z1 - w3 - w4 + 2*z1 <> 0 and a*b + b*d <> 0
3 2 2 2
and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c*z1 + a *b*c*w1 + a *c *w3
2 2 2 2 2
- 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 + a*b*c*d*w1 - 2*a*b*c*w3
2
- 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 - a*c*d*w2 + 2*a*c*w3 + a*c*w4
- 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4
- d*w2 + w3 + w4 - 2*z1 = 0,
2 2
{a *c*w5 - a *c*z1 + a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5
- a*z1 + d*w5 - d*z1 - w3 - w4 + 2*z1}},
2
{a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 - 2*b*w4 + 4*b*z1 + w3 + w4
- 2*z1 <> 0 and a*b <> 0 and a <> 0 and d = 0,
2
{(a*c*d - 2*b*d + d)*z4 - (a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3
- 2*b*w4 + 4*b*z1 + w3 + w4 - 2*z1)}},
2
{a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4 <> 0 and a*b <> 0 and a <> 0 and d <> 0
and a*c - 2*b + 1 = 0 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0
and a + d = 0,
2
{(a*c - 2*b + 1)*z3 + (a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4)}},
2
{a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1
<> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a*b + b*d = 0,
2
{(a*b + b*d)*z2 + (a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2
+ w3 + w4 - 2*z1)}},
2
{a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 <> 0 and a <> 0 and a*b = 0
and d = 0,
{(a*c*d + d)*z4 - (a*b)*z2
2
- (a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1)}},
2
{a *z1 - a*w2 + w4 <> 0 and a <> 0 and d <> 0 and a*b = 0 and a*c + 1 = 0,
2
{(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4)}},
3 2
{a*b + b*d <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c*z1 + a *b*c*w1
2 2 2 2 2 2 2
+ a *c *w3 - 2*a *c *z1 + a *c*d*z1 - a *c*w2 + a *z1 + a*b*c*d*w1
2
- 2*a*b*c*w3 - 2*a*b*c*w4 + 4*a*b*c*z1 + a*b*w1 - a*c *d*w4 - a*c*d*w2
+ 2*a*c*w3 + a*c*w4 - 4*a*c*z1 + a*d*z1 - a*w2 + b*d*w1 - 2*b*w3 - 2*b*w4
2 2
+ 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 and a *c*w5 - a *c*z1
+ a*c*d*w5 - a*c*d*z1 - a*c*w3 - a*c*w4 + 2*a*c*z1 + a*w5 - a*z1 + d*w5
- d*z1 - w3 - w4 + 2*z1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
2
(a*b + b*d)*z2 + (a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2
+ w3 + w4 - 2*z1)}},
3 2
{a*b <> 0 and a*c - 2*b + 1 <> 0 and a <> 0 and d <> 0 and a *c*z1 + a *b*c*w1
2 2 2
- a *c*w2 - a *c*w5 + a *c*z1 - 2*a*b*c*w4 + 2*a*b*w5 - 2*a*b*z1 + 2*a*c*w4
- a*w5 + a*z1 - 2*b*w4 + w4 = 0 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0
and a + d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
2
(a*c - 2*b + 1)*z3 + (a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4)}},
{a*b <> 0 and a*w3 - 2*a*z1 - d*w4 <> 0 and a <> 0 and d <> 0 and a*c + 1 = 0
and a + d = 0,
{(a + d)*z3 - (a*w3 - 2*a*z1 - d*w4)}},
2
{a*b <> 0 and a*w5 - a*z1 - w3 - w4 + 2*z1 <> 0 and a <> 0 and a *z1 + a*b*w1
+ a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3 - 2*b*w4 + 4*b*z1 + w3 + w4 - 2*z1 = 0
and d = 0,
{d*z4 + (a*w5 - a*z1 - w3 - w4 + 2*z1)}},
3 2
{a*b <> 0 and a + d <> 0 and a <> 0 and d <> 0 and a *z1 + 2*a *b*c*w3
2 2 2 2
- 4*a *b*c*z1 + a *b*w1 + a *d*z1 - a *w2 - 2*a*b*c*d*w4 + a*b*d*w1
- 2*a*b*w4 - a*d*w2 + a*w4 - 2*b*d*w4 + d*w4 = 0 and
2 2 2 2
a *c*w3 - 2*a *c*z1 + a *w5 - a *z1 - a*c*d*w4 + a*d*w5 - a*d*z1 - a*w4 - d*w4
= 0 and a*c + 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
(a + d)*z3 - (a*w3 - 2*a*z1 - d*w4)}},
2
{a*b <> 0 and a <> 0 and d <> 0 and a *z1 + a*b*w1 - a*w2 - 2*b*w4 + w4 = 0
and a*c - 2*b + 1 = 0 and a*c + 1 = 0 and a*w3 - 2*a*z1 - d*w4 = 0
and a + d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
z3 - (a*w5 - a*z1 - w4)}},
2
{a*b <> 0 and a <> 0 and a *z1 + a*b*w1 + a*c*w3 - 2*a*c*z1 - a*w2 - 2*b*w3
- 2*b*w4 + 4*b*z1 + w3 + w4 - 2*z1 = 0 and a*w5 - a*z1 - w3 - w4 + 2*z1 = 0
and d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(a*c*d + d)*z4 - (a*b)*z2
2
- (a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1),
z4 - (w5 - z1)}},
2 2
{a*c*d *z1 + b*c*d *w1 - 2*b*c*d*w3 + 4*b*c*d*z1 - 2*b*d*w5 + 2*b*d*z1 + 2*b*w3
2 2 2
- 4*b*z1 - c*d *w2 - c*d *w5 + c*d *z1 + 2*c*d*w3 - 4*c*d*z1 + d*w5 - d*z1
- w3 + 2*z1 <> 0 and a <> 0 and b*d <> 0 and 2*b + c*d - 1 <> 0 and d <> 0
2
and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0 and c*d - 1 = 0,
2 2 2 2
{(2*b*c*d - 2*b + c *d - 2*c*d + 1)*z3 + (a*c*d *z1 + b*c*d *w1 - 2*b*c*d*w3
2 2
+ 4*b*c*d*z1 - 2*b*d*w5 + 2*b*d*z1 + 2*b*w3 - 4*b*z1 - c*d *w2 - c*d *w5
2
+ c*d *z1 + 2*c*d*w3 - 4*c*d*z1 + d*w5 - d*z1 - w3 + 2*z1)}},
{a*c*d*z1 - a*z1 + b*c*d*w1 - 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1
2 2 2 2 2 2
- c *d *w5 + c *d *z1 + c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1
- c*w3 + 2*c*z1 + w2 - w5 + z1 <> 0 and a <> 0 and 2*b + c*d - 1 <> 0
2
and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0,
{a*c*d*z1 - a*z1 + b*c*d*w1 - 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1
2 2 2 2 2 2
- c *d *w5 + c *d *z1 + c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1
- c*w3 + 2*c*z1 + w2 - w5 + z1}},
{a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 <> 0 and a*c + 1 <> 0 and a <> 0
and d <> 0 and
2
a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*b + b*d = 0 and a*c - 2*b + 1 = 0,
{(a*c - 2*b + 1)*z2 - (a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2)}},
{a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1 <> 0 and a*c + 1 <> 0 and a <> 0
and d <> 0 and
2
a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*b + b*d = 0 and a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0
and a*c - 2*b + 1 = 0 and b = 0,
{b*z2 + (a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1)}},
2 2
{a*c - 2*b + 1 <> 0 and a*c + 1 <> 0 and a <> 0 and d <> 0 and a *c *w5
2 2 2 2
- a *c *z1 + a *c*z1 + a*b*c*w1 - 2*a*b*c*w5 + 2*a*b*c*z1 - a*c *w4 - a*c*w2
+ 2*a*c*w5 - 2*a*c*z1 + a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1
= 0 and
2
a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*b + b*d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
(a*c - 2*b + 1)*z2 - (a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2)}},
{a*c + 1 <> 0 and a <> 0 and b <> 0 and d <> 0 and
2
a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*b + b*d = 0 and a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0
and a*c - 2*b + 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4),
b*z2 + (a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1)}},
{a*c + 1 <> 0 and a <> 0 and d <> 0 and
2
a *z1 + a*c*w3 - 2*a*c*z1 + a*d*z1 - a*w2 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*b + b*d = 0 and a*c*w1 + 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0
and a*c*w5 - a*c*z1 + a*z1 - c*w4 - w2 + w5 - z1 = 0 and a*c - 2*b + 1 = 0
and b = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
2
(a*c + 1)*z3 + (a*b)*z2 + (a *z1 - a*w2 + w4)}},
{a*d*z1 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1
<> 0 and b*d <> 0 and d <> 0 and a = 0,
{(2*a*b + a*c*d - a)*z4 + (a*d*z1 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4
- d*w2 + w3 + w4 - 2*z1)}},
{a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1 <> 0 and a <> 0
2
and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0
and 2*b + c*d - 1 = 0 and c*d - 1 = 0,
{(2*b + c*d - 1)*z3 + (a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1)}}
,
{a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 <> 0 and d <> 0 and a = 0 and b*d = 0,
{(a*c*d - a)*z4 + (b*d)*z2 + (a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1)}},
2
{a*d*z1 - d*w2 + w3 - 2*z1 <> 0 and a <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0
and a*b = 0 and a*c + 1 = 0 and b*d = 0 and c*d - 1 = 0,
{(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1)}},
{2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 <> 0 and a <> 0 and c*d - 1 <> 0
2
and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0
and 2*b + c*d - 1 = 0,
{(2*b + c*d - 1)*z2 + (2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2)}},
{2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 <> 0 and a <> 0
2
and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 and a*b = 0
and 2*b - 1 = 0 and d = 0,
{(2*c*d)*z4 - (2*b - 1)*z2 - (2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2)}},
{2*a*z1 - 2*c*w4 + w1 - 2*w2 <> 0 and d <> 0
and a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 and a = 0 and b*d = 0
and 2*b - 1 = 0,
{(2*a*c)*z4 + (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2)}},
{2*a*z1 - 2*c*w4 + w1 - 2*w2 <> 0 and a = 0 and 2*b - 1 = 0 and d = 0
and w3 + w4 - 2*z1 = 0,
{(2*a*c)*z4 + (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2)}},
2
{2*a*z1 + w1 - 2*w2 <> 0 and a <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0
and a*b = 0 and a*c + 1 = 0 and a*d*z1 - d*w2 + w3 - 2*z1 = 0 and b*d = 0
and 2*b - 1 = 0 and c*d - 1 = 0 and c = 0,
{(2*c)*z3 + (2*b - 1)*z2 + (2*a*z1 + w1 - 2*w2)}},
{a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 <> 0 and a <> 0
2
and 2*b - 1 <> 0 and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0
and a*b = 0 and d = 0,
{(c*d)*z4 - (a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1)}},
{a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 <> 0 and 2*b - 1 <> 0
and d <> 0 and a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0 and a = 0
and b*d = 0,
{(a*c)*z4 + (a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1)}},
{a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 <> 0 and 2*b - 1 <> 0
and a = 0 and d = 0 and w3 + w4 - 2*z1 = 0,
{(a*c)*z4 + (a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1)}},
{a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - w2 + w5 - z1 <> 0 and a <> 0 and 2*b - 1 <> 0
2
and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0
and a*d*z1 - d*w2 + w3 - 2*z1 = 0 and b*d = 0 and c*d - 1 = 0 and c = 0,
{c*z3 + (a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - w2 + w5 - z1)}},
{a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 <> 0 and a <> 0
2
and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0 and 2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0
and 2*b + c*d - 1 = 0 and b = 0,
{b*z2 + (a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1)}},
{a <> 0 and b*d <> 0 and 2*b + c*d - 1 <> 0 and d <> 0
2 2 2
and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c*d *z1 + b*c*d *w1 - 2*b*c*d*w3
2 2
+ 4*b*c*d*z1 - 2*b*d*w5 + 2*b*d*z1 + 2*b*w3 - 4*b*z1 - c*d *w2 - c*d *w5
2
+ c*d *z1 + 2*c*d*w3 - 4*c*d*z1 + d*w5 - d*z1 - w3 + 2*z1 = 0 and a*c + 1 = 0
and c*d - 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1),
(2*b + c*d - 1)*z3 + (a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1)}}
,
2
{a <> 0 and b*d <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0 and a*d*z1 + b*d*w1 - 2*b*w3 + 4*b*z1 - d*w2 + w3 - 2*z1 = 0
and 2*b + c*d - 1 = 0 and c*d - 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1),
z3 + (d*w5 - d*z1 - w3 + 2*z1)}},
{a <> 0 and 2*b + c*d - 1 <> 0 and c*d - 1 <> 0 and d <> 0
2
and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c*d*z1 - a*z1 + b*c*d*w1
2 2 2 2
- 2*b*c*d*w5 + 2*b*c*d*z1 - b*w1 + 2*b*w5 - 2*b*z1 - c *d *w5 + c *d *z1
2 2
+ c *d*w3 - 2*c *d*z1 - c*d*w2 + 2*c*d*w5 - 2*c*d*z1 - c*w3 + 2*c*z1 + w2
- w5 + z1 = 0 and a*c + 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1),
(2*b + c*d - 1)*z2 + (2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2)}},
2
{a <> 0 and 2*b - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0 and a*d*z1 - d*w2 + w3 - 2*z1 = 0
and a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - w2 + w5 - z1 = 0 and b*d = 0
and c*d - 1 = 0 and c = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(2*c)*z3 + (2*b - 1)*z2 + (2*a*z1 + w1 - 2*w2)}},
{a <> 0 and 2*b - 1 <> 0
2
and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 and a*b = 0
and a*z1 + b*w1 - 2*b*w5 + 2*b*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 = 0
and d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(2*c*d)*z4 - (2*b - 1)*z2 - (2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2),
z4 - (w5 - z1)}},
2
{a <> 0 and b <> 0 and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0
and a*b = 0 and a*c + 1 = 0
and 2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0 and 2*b + c*d - 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1),
b*z2 + (a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1)}},
2
{a <> 0 and c*d - 1 <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0 and 2*a*z1 - c*d*w1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0
and a*z1 - c*d*w5 + c*d*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1 = 0
and 2*b + c*d - 1 = 0 and b = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(c*d - 1)*z3 + (b*d)*z2 + (a*d*z1 - d*w2 + w3 - 2*z1)}},
2
{a <> 0 and c <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0
and a*c + 1 = 0 and a*d*z1 - d*w2 + w3 - 2*z1 = 0 and b*d = 0 and c*d - 1 = 0
,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
(2*c)*z3 + (2*b - 1)*z2 + (2*a*z1 + w1 - 2*w2),
z2 - (w1 - 2*w5 + 2*z1)}},
2
{a <> 0 and d <> 0 and a *z1 - a*w2 + w4 = 0 and a*b = 0 and a*c + 1 = 0
and a*d*z1 - d*w2 + w3 - 2*z1 = 0 and 2*a*z1 + w1 - 2*w2 = 0 and b*d = 0
and 2*b - 1 = 0 and c*d - 1 = 0 and c = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
z4 - (w5 - z1),
c*z3 + b*z2 + (a*z1 - w2 + w5 - z1)}},
2
{a <> 0 and a *z1 + a*c*w3 - 2*a*c*z1 - a*w2 + w3 + w4 - 2*z1 = 0 and a*b = 0
and 2*a*z1 + 2*c*w3 - 4*c*z1 + w1 - 2*w2 = 0 and 2*b - 1 = 0 and d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(c*d)*z4 - b*z2 - (a*z1 + c*w3 - 2*c*z1 - w2 + w5 - z1),
z4 - (w5 - z1)}},
{b*d <> 0 and d*w5 - d*z1 - w3 - w4 + 2*z1 <> 0 and d <> 0 and
a*d*z1 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1
= 0 and a = 0,
{a*z4 + (d*w5 - d*z1 - w3 - w4 + 2*z1)}},
{b*d <> 0 and d <> 0 and
a*d*z1 + b*d*w1 - 2*b*w3 - 2*b*w4 + 4*b*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1
= 0 and a = 0 and d*w5 - d*z1 - w3 - w4 + 2*z1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(a*c*d - a)*z4 + (b*d)*z2 + (a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1),
z4 - (w5 - z1)}},
{2*b - 1 <> 0 and d <> 0 and a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 = 0 and a = 0
and b*d = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(2*a*c)*z4 + (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2),
z4 - (w5 - z1)}},
{2*b - 1 <> 0 and a*z1 + b*w1 - 2*b*w5 + 2*b*z1 - c*w4 - w2 + w5 - z1 = 0
and a = 0 and d = 0 and w3 + w4 - 2*z1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(2*a*c)*z4 + (2*b - 1)*z2 + (2*a*z1 - 2*c*w4 + w1 - 2*w2),
z4 - (w5 - z1)}},
{d <> 0 and a*d*z1 - c*d*w4 - d*w2 + w3 + w4 - 2*z1 = 0
and 2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0 and a = 0 and b*d = 0 and 2*b - 1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(a*c)*z4 + b*z2 + (a*z1 - c*w4 - w2 + w5 - z1),
z4 - (w5 - z1)}},
{w3 + w4 - 2*z1 <> 0 and a = 0 and d = 0,
{(a + d)*z4 - (w3 + w4 - 2*z1)}},
{2*a*z1 - 2*c*w4 + w1 - 2*w2 = 0 and a = 0 and 2*b - 1 = 0 and d = 0
and w3 + w4 - 2*z1 = 0,
{z4 + c*z3 + b*z2 + (a*z1 - w2),
a*z4 - z3 - w4,
d*z4 + z3 - (w3 - 2*z1),
2*z4 + z2 - w1,
(a*c)*z4 + b*z2 + (a*z1 - c*w4 - w2 + w5 - z1),
z4 - (w5 - z1)}}}
torder oo;
{{z4,z3,z2,z2},revgradlex}
% Two dimensional transportation problem
oo := torder({x33,x32,x31,x23,x22,x21,x13,x12,x11},lex);
oo := {{},lex}
gsys({x11+x12+x13-a1,x11+x21+x31-b1,x12+x22+x32-b2,x13+x23+x33-b3,
x21+x22+x23-a2,x31+x32+x33-a3});
{{a1 + a2 + a3 - b1 - b2 - b3 <> 0,
{a1 + a2 + a3 - b1 - b2 - b3}},
{a1 + a2 + a3 - b1 - b2 - b3 = 0,
{x33 + x32 + x31 - a3,
x33 + x23 + x13 - b3,
x32 + x22 + x12 - b2,
x31 + x21 + x11 - b1,
x23 + x22 + x21 - a2,
x13 + x12 + x11 - a1}}}
torder oo;
{{x33,x32,x31,x23,x22,x21,x13,x12,x11},lex}
% Thomas Weis's Example 1
oo := torder({x,y,z},lex);
oo := {{},lex}
gsys({z*y*x-b*y*x-b*z*x+b**2*x-b*z*y+b**2*y+b**2*z-(n3+b**3),
z*y*x-a*y*x-a*z*x+a**2*x-a*z*y+a**2*y+a**2*z-(n3+a**3),
z*y*x-n1});
3 3
{{a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3 <> 0 and a - b <> 0
2 2
and a *b - a*b = 0,
2 2 2 2 2 2
{(a *b - a*b )*x + (a *b - a*b )*y + (a *b - a*b )*z
3 3
- (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)}},
3
{a - n1 + n3 <> 0 and a - b = 0 and a = 0,
2 2 2 3
{a*x*y + a*x*z - a *x + a*y*z - a *y - a *z + (a - n1 + n3)}},
2 2
{a *b - a*b <> 0 and a - b <> 0,
2 2 2 3
{x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3),
2 2 2 3
x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3),
x*y*z - n1,
2 2 2 2
(a - b)*x*y + (a - b)*x*z - (a - b )*x + (a - b)*y*z - (a - b )*y
2 2 3 3
- (a - b )*z + (a - b ),
2 2 2 2 2 2
(a *b - a*b )*x + (a *b - a*b )*y + (a *b - a*b )*z
3 3
- (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3),
2 2 2 2 2
(a *b - a*b )*y + (a *b - a*b )*y*z
3 3 2 2 2
- (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*y + (a *b - a*b )*z
3 3
- (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*z
3 2 2 3 2 2 2 2
+ (a *b - a *b + a *n1 - a *n3 - b *n1 + b *n3),
2 2 3 3 3 2
(a *b - a*b )*z - (a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3)*z
3 2 2 3 2 2 2 2 2 2
+ (a *b - a *b + a *n1 - a *n3 - b *n1 + b *n3)*z - (a *b*n1 - a*b *n1)}},
3 3 2 2
{a - b <> 0 and a *b - a*b + a*n1 - a*n3 - b*n1 + b*n3 = 0 and a *b - a*b = 0
,
2 2 2 3
{x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3),
2 2 2 3
x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3),
x*y*z - n1,
2 2 2 2
(a - b)*x*y + (a - b)*x*z - (a - b )*x + (a - b)*y*z - (a - b )*y
2 2 3 3
- (a - b )*z + (a - b ),
2 2 2 2 2 2 2 2
(a - b)*x*z - (a - b )*x*z + (a *b - a*b )*x + (a - b)*y*z - (a - b )*y*z
2 2 2 2 2 3 2 2 3
+ (a *b - a*b )*y - (a - b )*z + (a + a *b - a*b - b )*z
3 3
- (a *b - a*b - a*n3 + b*n3),
2 2 2 2 2 2 2 2 2 2 2
(a - b)*y *z - (a - b )*y *z + (a *b - a*b )*y - (a - b )*y*z
3 2 2 3 3 3
+ (a + a *b - a*b - b )*y*z - (a *b - a*b - a*n3 + b*n3)*y
2 2 2 3 3
+ (a *b - a*b )*z - (a *b - a*b - a*n3 + b*n3)*z
3 2 2 3 2 2
+ (a *b - a *b - a *n3 + b *n3)}},
{a <> 0 and a - b = 0,
2 2 2 3
{x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3),
2 2 2 3
x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3),
x*y*z - n1,
2 2 2 3
a*x*y + a*x*z - a *x + a*y*z - a *y - a *z + (a - n1 + n3),
2 2 2 2 2 2 3
a*x*z - a *x*z + a*y*z - a *y*z - a *z + (a - n1 + n3)*z + a*n1,
2 2 2 2 2 2 3
a*y *z - a *y *z - a *y*z + (a - n1 + n3)*y*z + (a*n1)*y + (a*n1)*z
2
- a *n1}},
3
{a - n1 + n3 = 0 and a - b = 0 and a = 0,
2 2 2 3
{x*y*z - a*x*y - a*x*z + a *x - a*y*z + a *y + a *z - (a + n3),
2 2 2 3
x*y*z - b*x*y - b*x*z + b *x - b*y*z + b *y + b *z - (b + n3),
x*y*z - n1}}}
torder oo;
{{x,y,z},lex}
% Thomas Weis's Example 2
oo := torder({z,y,x,w},lex);
oo := {{},lex}
gsys({w*x*y*z-x*y*z-w*y*z+y*z-w*x*z+x*z+w*z-z-w*x*y+x*y+w*y-
y+w*x-x-w-(b-1),
w*x*y*z-2*x*y*z-2*w*y*z+4*y*z-2*w*x*z+4*x*z+4*w*z-8*z-2*w*x*y+4x*y+
4*w*y-8*y+4*w*x-8*x-8*w-(c-16),
w*x*y*z-a,z+y+x+w-v});
{{true,
{z*y*x*w - z*y*x - z*y*w + z*y - z*x*w + z*x + z*w - z - y*x*w + y*x + y*w - y
+ x*w - x - w - (b - 1),
z*y*x*w - 2*z*y*x - 2*z*y*w + 4*z*y - 2*z*x*w + 4*z*x + 4*z*w - 8*z - 2*y*x*w
+ 4*y*x + 4*y*w - 8*y + 4*x*w - 8*x - 8*w - (c - 16),
z*y*x*w - a,
z + y + x + w - v,
2 2 2 2 2
y *x + y *w - 3*y + y*x + 2*y*x*w - (v + 3)*y*x + y*w - (v + 3)*y*w
2 2 2 2
+ (3*v)*y + x *w - 3*x + x*w - (v + 3)*x*w + (3*v)*x - 3*w + (3*v)*w
+ (b - c - 7*v + 15),
2 2 2
2*y + 2*y*x + 2*y*w - (2*v)*y + 2*x + 2*x*w - (2*v)*x + 2*w - (2*v)*w
+ (a - 2*b + c + 6*v - 14),
3 2 2 2
2*x + 2*x *w - (2*v)*x + 2*x*w - (2*v)*x*w + (a - 2*b + c + 6*v - 14)*x
3 2
+ 2*w - (2*v)*w + (a - 2*b + c + 6*v - 14)*w - (3*a - 4*b + c + 4*v - 12),
4 3 2
2*w - (2*v)*w + (a - 2*b + c + 6*v - 14)*w - (3*a - 4*b + c + 4*v - 12)*w
+ 2*a}}}
torder oo;
{{z,y,x,w},lex}
end;
Time for test: 12317 ms, plus GC time: 91 ms