Tue Feb 10 12:26:42 2004 run on Linux
% Author H.-G. Graebe | Univ. Leipzig | Version 28.6.1995
% graebe@informatik.uni-leipzig.de
COMMENT
This is an example session demonstrating and testing the facilities
offered by the commutative algebra package CALI.
END COMMENT;
algebraic;
on echo;
off nat;
% To make it easier to compare differing output.
showtime;
Time: 0 ms
comment
####################################
### ###
### Introductory Examples ###
### ###
####################################
end comment;
% Example 1 : Generating ideals of affine and projective points.
vars:={t,x,y,z};
vars := {t,x,y,z}$
setring(vars,degreeorder vars,revlex);
{{t,x,y,z},{{1,1,1,1}},revlex,{1,1,1,1}}$
mm:=mat((1,1,1,1),(3,2,3,1),(2,1,3,2));
mm := mat((1,1,1,1),(3,2,3,1),(2,1,3,2))$
% The ideal with zero set at the point in A^4 with coordinates
% equal to the row vectors of mm :
setideal(m1,affine_points mm);
{z**2 - 3*z + 2,
y**2 - 4*y + 3,
t - y + z - 1,
2*x - y + 2*z - 3,
y*z - y - 3*z + 3}$
% All parameters are as they should be :
dim m1;
0$
degree m1;
3$
groebfactor m1;
{{z - 2,y - 3,t - 2,x - 1},
{z - 1,t - 3,y - 3,x - 2},
{z - 1,t - 1,y - 1,x - 1}}$
resolve m1$
bettinumbers m1;
{1,5,9,7,2}$
% The ideal with zero set at the point in P^3 with homogeneous
% coordinates equal to the row vectors of mm :
setideal(m2,proj_points mm);
{2*y**2 - 2*x*z - 7*y*z + 7*z**2,
3*t - 2*x - 2*y + z,
2*x**2 - 4*x*z - y*z + 3*z**2,
2*x*y - 4*x*z - 3*y*z + 5*z**2}$
% All parameters as they should be ?
dim m2;
1$
degree m2;
3$
groebfactor m2;
{{2*y**2 - 2*x*z - 7*y*z + 7*z**2,
3*t - 2*x - 2*y + z,
2*x**2 - 4*x*z - y*z + 3*z**2,
2*x*y - 4*x*z - 3*y*z + 5*z**2}}$
% It seems to be prime ?
isprime m2;
no$
% Not, of course, but it is known to be unmixed.
% Hence we can use
easyprimarydecomposition m2;
{{{x - 2*z,y - 3*z,t - 3*z},
{y - 3*z,x - 2*z,t - 3*z}},
{{x - z,y - z,t - z},
{y - z,x - z,t - z}},
{{2*x - z,2*y - 3*z,t - z},
{2*y - 3*z,2*x - z,t - z}}}$
% Example 2 :
% The affine monomial curve with generic point (t^7,t^9,t^10).
setideal(m,affine_monomial_curve({7,9,10},{x,y,z}));
{x**3*y - z**3,
x**4 - y**2*z,
y**3 - x*z**2}$
% The base ring was changed as side effect :
getring();
{{x,y,z},{{7,9,10}},revlex,{7,9,10}}$
vars:=first getring m;
vars := {x,y,z}$
% Some advanced commutative algebra :
% The analytic spread of m.
analytic_spread m;
3$
% The Rees ring Rees_R(vars) over R=S/m.
rees:=blowup(m,vars,{u,v,w});
rees := {u**2*v*x - w**3,
u*v*x**2 - w**2*z,
v*x**3 - w*z**2,
u**3*x - v**2*w,
u**2*x**2 - v*w*y,
u*x**3 - w*y**2,
- u*w**2 + v**3,
v**2*y - w**2*x,
v*y**2 - w*x*z,
v*z - w*y,
u*z - w*x,
u*y - v*x,
x**3*y - z**3,
x**4 - y**2*z,
- x*z**2 + y**3}$
% It is multihomogeneous wrt. the degree vectors, constructed during
% blow up. Lets compute weighted Hilbert series :
setideal(rees,rees)$
weights:=second getring();
weights := {{0,0,0,7,9,10},{7,9,10,0,0,0}}$
weightedhilbertseries(gbasis rees,weights);
( - x**29*y + x**29 - x**20*y + x**20 - x**19*y**11 + x**19*y**10 - x**19*y + x
**19 - x**18*y + x**18 - x**10*y**11 + x**10*y**10 - x**10*y + x**10 - x**9*y**
21 + x**9*y**20 - x**9*y**11 + x**9*y**9 - x**9*y + x**9 + y**23 - y**22 + y**16
- y**15 + y**14 - y**11 + y**9 - y**8 + y**7 - y + 1)/(x**7*y - x**7 - y + 1)$
% gr_R(vars), the associated graded ring of the irrelevant ideal
% over R. The short way.
interreduce sub(x=0,y=0,z=0,rees);
{w**3,v**2*w, - u*w**2 + v**3}$
% The long (and more general) way. Gives the result in another
% embedding.
% Restore the base ring, since it was changed by blowup as a side
% effect.
setring getring m$
assgrad(m,vars,{u,v,w});
{x,
y,
z,
w**3,
v**2*w,
- u*w**2 + v**3}$
% Comparing the Rees algebra and the symmetric algebra of M :
setring getring m$
setideal(rees,blowup({},m,{a,b,c}));
{ - y**2*a + z**2*b + x**3*c,
x*a - y*b - z*c,
- y**2*a**2 + z**2*a*b + x**2*y*b*c + x**2*z*c**2,
- y**2*a**3 + z**2*a**2*b + x*y**2*b**2*c + 2*x*y*z*b*c**2 + x*z**2*c**3,
- y**2*a**4 + z**2*a**3*b + y**3*b**3*c + 3*y**2*z*b**2*c**2 + 3*y*z**2*b*c**3
+ z**3*c**4}$
% Lets test weighted Hilbert series once more :
weights:=second getring();
weights := {{0,0,0,30,28,27},{7,9,10,0,0,0}}$
weightedhilbertseries(gbasis rees,weights);
(x**58*y**27 + x**30*y**25 - x**30*y**18 - x**30*y**7 - x**28*y**27 + 1)/(x**85*
y**26 - x**85*y**19 - x**85*y**17 - x**85*y**16 + x**85*y**10 + x**85*y**9 + x**
85*y**7 - x**85 - x**58*y**26 + x**58*y**19 + x**58*y**17 + x**58*y**16 - x**58*
y**10 - x**58*y**9 - x**58*y**7 + x**58 - x**57*y**26 + x**57*y**19 + x**57*y**
17 + x**57*y**16 - x**57*y**10 - x**57*y**9 - x**57*y**7 + x**57 - x**55*y**26 +
x**55*y**19 + x**55*y**17 + x**55*y**16 - x**55*y**10 - x**55*y**9 - x**55*y**7
+ x**55 + x**30*y**26 - x**30*y**19 - x**30*y**17 - x**30*y**16 + x**30*y**10 +
x**30*y**9 + x**30*y**7 - x**30 + x**28*y**26 - x**28*y**19 - x**28*y**17 - x**
28*y**16 + x**28*y**10 + x**28*y**9 + x**28*y**7 - x**28 + x**27*y**26 - x**27*y
**19 - x**27*y**17 - x**27*y**16 + x**27*y**10 + x**27*y**9 + x**27*y**7 - x**27
- y**26 + y**19 + y**17 + y**16 - y**10 - y**9 - y**7 + 1)$
% The symmetric algebra :
setring getring m$
setideal(sym,sym(m,{a,b,c}));
{y**2*a - z**2*b - x**3*c,
x*a - y*b - z*c}$
modequalp(rees,sym);
yes$
% Symbolic powers :
setring getring m$
setideal(m2,idealpower(m,2));
{x**6*y**2 - 2*x**3*y*z**3 + z**6,
x**8 - 2*x**4*y**2*z + y**4*z**2,
y**6 - 2*x*y**3*z**2 + x**2*z**4,
x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4,
x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5,
x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3}$
% Let's compute a second symbolic power :
setideal(m3,symbolic_power(m,2));
{x**6*y**2 - 2*x**3*y*z**3 + z**6,
x**8 - 2*x**4*y**2*z + y**4*z**2,
y**6 - 2*x*y**3*z**2 + x**2*z**4,
x**2*y**5 + x**7*z - 3*x**3*y**2*z**2 + y*z**5,
x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4,
x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5,
x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3}$
% It is different from the ordinary second power.
% Hence m2 has a trivial component.
modequalp(m2,m3);
no$
% Test x for non zero divisor property :
nzdp(x,m2);
no$
nzdp(x,m3);
yes$
% Here is the primary decomposition :
pd:=primarydecomposition m2;
pd := {{{x**8 - 2*x**4*y**2*z + y**4*z**2,
x**6*y**2 - 2*x**3*y*z**3 + z**6,
x**2*z**4 - 2*x*y**3*z**2 + y**6,
x**7*z - 3*x**3*y**2*z**2 + x**2*y**5 + y*z**5,
x**7*y - x**4*z**3 - x**3*y**3*z + y**2*z**4,
- x**4*y*z**2 + x**3*y**4 + x*z**5 - y**3*z**3,
- x**5*z**2 + x**4*y**3 + x*y**2*z**3 - y**5*z},
{ - x*z**2 + y**3,
x**4 - y**2*z,
x**3*y - z**3}},
{{z**2,
x**6*y**2,
y**6,
x**2*y**5*z,
x**3*y**4,
x**4*(x**4 - 2*y**2*z),
x**3*y*(x**4 - y**2*z),
y**3*(x**4 - y**2*z)},
{x,z,y}}}$
% Compare the result with m2 :
setideal(m4,matintersect(first first pd, first second pd));
{y**6 - 2*x*y**3*z**2 + x**2*z**4,
x**6*y**2 - 2*x**3*y*z**3 + z**6,
x**8 - 2*x**4*y**2*z + y**4*z**2,
x**2*y**5*z + x**7*z**2 - 3*x**3*y**2*z**3 + y*z**6,
x**4*y**3 - y**5*z - x**5*z**2 + x*y**2*z**3,
x**3*y**4 - x**4*y*z**2 - y**3*z**3 + x*z**5,
x**7*y - x**3*y**3*z - x**4*z**3 + y**2*z**4}$
modequalp(m2,m4);
yes$
% Compare the result with m3 :
setideal(m4,first first pd)$
modequalp(m3,m4);
yes$
% The trivial component can also be removed with a stable
% quotient computation :
setideal(m5,matstabquot(m2,vars))$
modequalp(m3,m5);
yes$
% Example 3 : The Macaulay curve.
setideal(m,proj_monomial_curve({0,1,3,4},{w,x,y,z}));
{x**3 - w**2*y,
w*y**2 - x**2*z,
y**3 - x*z**2,
x*y - w*z}$
vars:=first getring();
vars := {w,x,y,z}$
gbasis m;
{x**3 - w**2*y,
w*y**2 - x**2*z,
y**3 - x*z**2,
x*y - w*z}$
% Test whether m is prime :
isprime m;
yes$
% A resolution of m :
resolve m;
{
mat((x**3 - w**2*y),(w*y**2 - x**2*z),(y**3 - x*z**2),(x*y - w*z))$
,
mat((y,w,0, - x**2),(z,x,0, - w*y),(0, - y,w, - x*z),(0, - z,x, - y**2))$
,
mat((z, - y, - x,w))$
,
mat((0))$
}$
% m has depth = 1 as can be seen from the
gradedbettinumbers m;
{{0},{2,3,3,3},{4,4,4,4},{5}}$
% Another way to see the non perfectness of m :
hilbertseries m;
( - w**3 + 2*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$
% Just a third approach. Divide out a parameter system :
ps:=for i:=1:2 collect random_linear_form(vars,1000);
ps := {927*w + 880*x + 292*y + 9*z, - 819*w + 224*x - 572*y - 205*z}$
setideal(m1,matsum(m,ps))$
% dim should be zero and degree > degree m = 4.
% A Gbasis for m1 is computed automatically.
dim m1;
0$
degree m1;
5$
% The projections of m on the coord. hyperplanes.
for each x in vars collect eliminate(m,{x});
{{ - x*z**2 + y**3},
{ - w*z**3 + y**4},
{ - w**3*z + x**4},
{ - w**2*y + x**3}}$
% Example 4 : Two submodules of S^4.
% Get the stored result of the earlier computation.
r:=resolve m$
% See whether cali!=degrees contains a relict from earlier
% computations.
getdegrees();
{}$
% Introduce the 2nd and 3rd syzygy module as new modules.
% Both are submodules in S^4.
setmodule(m1,second r)$
setmodule(m2,third r)$
% The second is already a gbasis.
setgbasis m2;
mat((z, - y, - x,w))$
getleadterms m1;
mat((0,x**3,0,0),(0,0,x**3,0),(0,0,w**2*y,0),(0,0,w*y**2,0),(0,0,w*x,0),(0,0,0,x
**2),(0,0,0,w*y),(0,0,0,x*z),(0,0,0,y**2))$
getleadterms m2;
mat((0,0,0,w))$
% Since rk(F/M)=rk(F/in(M)), they have ranks 1 resp. 3.
dim m1;
4$
indepvarsets m1;
{{w,x,y,z}}$
% Its intersection is zero :
matintersect(m1,m2);
mat((0,0,0,0))$
% Its sum :
setmodule(m3,matsum(m1,m2));
mat(( - y, - w,0,x**2),(0,y, - w,x*z),(0,z, - x,y**2),(z, - y, - x,w),( - y*z -
z,y**2 - x,x*y,0))$
dim m3;
3$
% Hence it has a nontrivial annihilator :
annihilator m3;
{w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}$
% One can compute isolated primes and primary decomposition also for
% modules. Let's do it, although being trivial here:
isolatedprimes m3;
{{w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}}$
primarydecomposition m3;
{{
mat((z*( - w*x*z - w*y*z + w*y - w*z + x**2*z + x*y*z - y**3 + y**2*z),w**2*y +
w**2*z + w*x*y*z + w*x*z**2 - w*x*z + w*y*z**2 - x**2*z**2 + x*y**2*z, - w**3,0)
,(z*( - w*y - w - x*z + y**2), - w*x + w*y*z - x**2*z + x*y**2,w**2*y,0),( - x*z
**2,w*y + x*y*z + x*z**2 - y**3,w*( - w + y**2),0),( - x*z**2,y*(w + x*z), - w**
2 + x**2*z,0),( - y*z, - (w*z + x*y),w*x,0),( - w*y**2 + x**2*z, - w**2*y + x**3
,0,0),(w*y*z - w*y + w*z - x**2*z + y**3 - y**2*z, - w**2 + w*x - w*y*z + x**2*y
+ x**2*z - x*y**2,0,0),( - w*y*z - w*z - y**3 + y**2*z, - w*x + w*y*z - x**2*z
+ x*y**2,x**3,0),(z*( - w*y - w + y**2), - w*x + w*y**2 + w*y*z + x*y**2,0,0),(
- z*(y + 1), - x + y**2,x*y,0),(z, - y, - x,w),(w**2*y*z + w**2*z - w*x*y + w*y
**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3,0,0,0),( - y, - w,0,x**2),(0,y, - w
,x*z),(0,z, - x,y**2))$
,
{w**2*y*z + w**2*z - w*x*y + w*y**3 - x**3*z - x**2*y*z - x**2*z**2 + x*y**3}}}$
% To get a meaningful Hilbert series make m1 homogeneous :
setdegrees {1,x,x,x};
{1,x,x,x}$
% Reevaluate m1 with the new column degrees.
setmodule(m1,m1)$
hilbertseries m1;
(w**7 - 5*w**6 + 8*w**5 - 2*w**4 - 5*w**3 + 3*w + 1)/(w**4 - 4*w**3 + 6*w**2 - 4
*w + 1)$
% Example 5 : From the MACAULAY manual (D.Bayer, M.Stillman).
% An elliptic curve on the Veronese in P^5.
rvars:={x,y,z}$
svars:={a,b,c,d,e,f}$
r:=setring(rvars,degreeorder rvars,revlex)$
s:=setring(svars,{for each x in svars collect 2},revlex)$
map:={s,r,{a=x^2,b=x*y,c=x*z,d=y^2,e=y*z,f=z^2}};
map := {{{a,
b,
c,
d,
e,
f},
{{2,2,2,2,2,2}},
revlex,
{1,1,1,1,1,1}},
{{x,y,z},
{{1,1,1}},
revlex,
{1,1,1}},
{a=x**2,
b=x*y,
c=x*z,
d=y**2,
e=y*z,
f=z**2}}$
preimage({y^2z-x^3-x*z^2},map);
{ - a*d + b**2,
- a*e + b*c,
- b*e + c*d,
- a*f + c**2,
- b*f + c*e,
- d*f + e**2,
a**2 + a*f - b*e,
a*b + b*f - d*e,
a*c + c*f - d*f}$
% Example 6 : The preimage under a rational map.
r:=setring({x,y},{},lex)$
s:=setring({t},{},lex)$
map:={r,s,{x=2t/(t^2+1),y=(t^2-1)/(t^2+1)}};
map := {{{x,y},{},lex,{1,1}},
{{t},{},lex,{1}},
{x=(2*t)/(t**2 + 1),y=(t**2 - 1)/(t**2 + 1)}}$
% The preimage of (0) is the equation of the circle :
ratpreimage({},map);
{x**2 + y**2 - 1}$
% The preimage of the point (t=3/2) :
ratpreimage({2t-3},map);
{13*x - 12,13*y - 5}$
% Example 7 : A zerodimensional ideal.
setring({x,y,z},{},lex)$
setideal(n,{x**2 + y + z - 3,x + y**2 + z - 3,x + y + z**2 - 3});
{x**2 + y + z - 3,x + y**2 + z - 3,x + y + z**2 - 3}$
% The groebner algorithm with factorization :
groebfactor n;
{{y - 1,z - 1,x - 1},
{y + 3,z + 3,x + 3},
{y - z,z**2 - 2,x + z - 1},
{z**2 - 2,x - z,y + z - 1},
{y + z - 1,z**2 - 2*z - 1,x + z - 1}}$
% Change the term order and reevaluate n :
setring({x,y,z},{{1,1,1}},revlex)$
setideal(n,n);
{x**2 + y + z - 3,y**2 + x + z - 3,z**2 + x + y - 3}$
% its primes :
zeroprimes n;
{{x - z,z**2 - 2,y + z - 1},
{x + z - 1,y + z - 1,z**2 - 2*z - 1},
{z - 1,x - 1,y - 1},
{z + 3,x + 3,y + 3},
{y - z,z**2 - 2,x + z - 1}}$
% a vector space basis of S/n :
getkbase n;
{1,x,x*y,x*y*z,x*z,y,y*z,z}$
% Example 8 : A modular computation. Since REDUCE has no multivariate
% factorizer, factorprimes has to be turned off !
on modular$
off factorprimes$
setmod 181;
1$
setideal(n1,n);
{x**2 + y + z + 178,y**2 + x + z + 178,z**2 + x + y + 178}$
zeroprimes n1;
{{y + 180*z,z**2 + 179,x + z + 180},
{x + z + 180,y + z + 180,z**2 + 179*z + 180},
{x + 180*z,z**2 + 179,y + z + 180},
{z + 180,x + 180,y + 180},
{z + 3,x + 3,y + 3}}$
setmod 7;
181$
setideal(n1,n);
{x**2 + y + z + 4,y**2 + x + z + 4,z**2 + x + y + 4}$
zeroprimes n1;
{{z + 6,x + 6,y + 6},
{x + 4,z + 4,y + 2},
{x + 4,z + 2,y + 4},
{z + 4,x + 2,y + 4},
{x + 3,z + 3,y + 3}}$
% Hence some of the primes glue together mod 7.
zeroprimarydecomposition n1;
{{{z + 6,x + 6,y + 6},
{z + 6,x + 6,y + 6}},
{{z + 4,y + 2,x + 4},
{x + 4,z + 4,y + 2}},
{{z + 2,y + 4,x + 4},
{x + 4,z + 2,y + 4}},
{{z + 4,x + 2,y + 4},
{z + 4,x + 2,y + 4}},
{{x**2 + y + z + 4,
x + y**2 + z + 4,
x + y + z**2 + 4,
3*(x + 5*y*z + 2*y + 2*z + 5),
x*z + 6*x + 3*y + 6*z + 1,
x*y + 6*x + 6*y + 3*z + 1},
{x + 3,z + 3,y + 3}}}$
off modular$
on factorprimes$
% Example 9 : Independent sets once more.
n:=10$
vars:=for i:=1:(2*n) collect mkid(x,i)$
setring(vars,{},lex)$
setideal(m,for j:=0:n collect
for i:=(j+1):(j+n) product mkid(x,i));
{x1*x2*x3*x4*x5*x6*x7*x8*x9*x10,
x2*x3*x4*x5*x6*x7*x8*x9*x10*x11,
x3*x4*x5*x6*x7*x8*x9*x10*x11*x12,
x4*x5*x6*x7*x8*x9*x10*x11*x12*x13,
x5*x6*x7*x8*x9*x10*x11*x12*x13*x14,
x6*x7*x8*x9*x10*x11*x12*x13*x14*x15,
x7*x8*x9*x10*x11*x12*x13*x14*x15*x16,
x8*x9*x10*x11*x12*x13*x14*x15*x16*x17,
x9*x10*x11*x12*x13*x14*x15*x16*x17*x18,
x10*x11*x12*x13*x14*x15*x16*x17*x18*x19,
x11*x12*x13*x14*x15*x16*x17*x18*x19*x20}$
setgbasis m$
indepvarsets m;
{{x2,x3,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x3,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x4,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x5,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x6,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x12,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x3,x4,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x4,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x13,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x4,x5,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x14,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x5,x6,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x15,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x6,x7,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x16,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x7,x8,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x17,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x8,x9,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x18,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x9,x10,x11,x12,x13,x14,x15,x16,x17,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x17,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x19,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x10,x11,x12,x13,x14,x15,x16,x17,x18,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x18,x20},
{x1,x2,x3,x4,x5,x6,x7,x8,x9,x11,x12,x13,x14,x15,x16,x17,x18,x19}}$
dim m;
18$
degree m;
55$
comment
####################################
### ###
### Local Standard Bases ###
### ###
####################################
end comment;
% Example 10 : An example from [ Alonso, Mora, Raimondo ]
vars := {z,x,y}$
r:=setring(vars,{},lex)$
setideal(m,{x^3+(x^2-y^2)*z+z^4,y^3+(x^2-y^2)*z-z^4});
{z**4 + z*x**2 - z*y**2 + x**3,
- z**4 + z*x**2 - z*y**2 + y**3}$
dim m;
1$
degree m;
12$
% 2 = codim m is the codimension of the curve m. The defining
% equations of the singular locus with their nilpotent structure :
singular_locus(m,2);
{x**3 - y**3 + 2*z**4,
x**3 + 2*x**2*z + y**3 - 2*y**2*z,
y*(8*x**3 + 3*x**2*y - 11*y**3 + 12*y*z**3),
y*(x**3 + 3*x**2*y + 2*x*y*z + y**3 - 2*y**2*z),
3*x**5 + 3*x**4*y + 22*x**4 + 18*x**3*y**2 + 16*x**3*y + 21*x**2*y**3 + 3*x*y**4
- 16*x*y**3 + 18*y**5 - 42*y**4*z - 22*y**4 + 24*y**3*z**2}$
groebfactor ws;
{{y,x,z},{81*x + 256,27*z - 64,81*y - 256}}$
% Hence this curve has two singular points :
% (x=y=z=0) and (y=-x=256/81,z=64/27)
% Let's find the brances of the curve through the origin.
% The first critical tropism is (-1,-1,-1).
off noetherian$
setring(vars,{{-1,-1,-1}},lex)$
setideal(m,m);
{z*x**2 - z*y**2 + x**3 + z**4,
z*x**2 - z*y**2 + y**3 - z**4}$
% Let's first test two different approaches, not fully
% integrated into the algebraic interface :
setideal(m1,homstbasis m);
{x**3 - y**3 + 2*z**4,
z*x**2 - z*y**2 + y**3 - z**4,
z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x,
x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4*y
**2,
6*z*y**5 + 2*x*y**5 - 2*y**6 - 4*z**6*x - 4*z**6*y - 2*z**5*x*y - 8*z**5*y**2 +
z**4*x**3 - 2*z**4*x*y**2 + 3*z**4*y**3}$
setideal(m2,lazystbasis m);
{x**3 - y**3 + 2*z**4,
z*x**2 - z*y**2 + y**3 - z**4,
z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x,
x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4*y
**2,
3*z*y**5 + x*y**5 - y**6 - 2*z**6*x - 2*z**6*y - z**5*x**2 - z**5*x*y - 3*z**5*y
**2 - z**4*x*y**2 + z**4*y**3}$
setgbasis m1$
setgbasis m2$
modequalp(m1,m2);
yes$
gbasis m;
{x**3 - y**3 + 2*z**4,
z*x**2 - z*y**2 + y**3 - z**4,
z*x*y**2 - z*y**3 - x*y**3 + 2*z**5 + z**4*x,
x**2*y**3 + x*y**4 + y**5 - 2*z**5*x - 2*z**5*y - z**4*x**2 - z**4*x*y - z**4*y
**2,
3*z*y**5 + x*y**5 - y**6 - 2*z**6*x - 2*z**6*y - z**5*x**2 - z**5*x*y - 3*z**5*y
**2 - z**4*x*y**2 + z**4*y**3}$
modequalp(m,m1);
yes$
dim m;
1$
degree m;
9$
% Find the tangent directions not in z-direction :
tangentcone m;
{x**3 - y**3,
z*x**2 - z*y**2 + y**3,
z*x*y**2 - z*y**3 - x*y**3,
x**2*y**3 + x*y**4 + y**5,
3*z*y**5 + x*y**5 - y**6}$
setideal(n,sub(z=1,ws));
{x**3 - y**3,
x**2 - y**2 + y**3,
x*y**2 - y**3 - x*y**3,
x**2*y**3 + x*y**4 + y**5,
3*y**5 + x*y**5 - y**6}$
setring r$
on noetherian$
setideal(n,n)$
degree n;
9$
% The points of n outside the origin.
matstabquot(n,{x,y});
{y**2 - 3*y + 3,x - y + 3}$
% Hence there are two branches x=z'*(a-3+x'),y=z'*(a+y'),z=z'
% with the algebraic number a : a^2-3a+3=0
% and the new equations for (z',x',y') :
setrules {a^2=>3a-3};
{a**2 => 3*a - 3}$
sub(x=z*(a-3+x),y=z*(a+y),m);
{z**3*(a**3 + 3*a**2*x - 9*a**2 + 3*a*x**2 - 16*a*x - 2*a*y + 21*a + x**3 - 8*x
**2 + 21*x - y**2 + z - 18),
z**3*(a**3 + 3*a**2*y + 2*a*x + 3*a*y**2 - 2*a*y - 6*a + x**2 - 6*x + y**3 - y**
2 - z + 9)}$
setideal(m1,matqquot(ws,z));
{x**3 + (3*a - 7)*x**2 - (5*a - 6)*x + y**3 + (3*a - 2)*y**2 + (5*a - 9)*y,
z - x**2 - (2*a - 6)*x - y**3 - (3*a - 1)*y**2 - (7*a - 9)*y}$
% This defines a loc. smooth system at the origin, since the
% jacobian at the origin of the gbasis is nonsingular :
off noetherian$
setring getring m;
{{z,x,y},{{-1,-1,-1}},lex,{1,1,1}}$
setideal(m1,m1);
{ - (5*a - 6)*x + (5*a - 9)*y + (3*a - 7)*x**2 + (3*a - 2)*y**2 + x**3 + y**3,
z - (2*a - 6)*x - (7*a - 9)*y - x**2 - (3*a - 1)*y**2 - y**3}$
gbasis m1;
{(5*a - 6)*x - (5*a - 9)*y - (3*a - 7)*x**2 - (3*a - 2)*y**2 - x**3 - y**3,
(5*a - 6)*z + 27*y + (9*a - 18)*x**2 - (18*a - 45)*y**2 - (2*a - 6)*x**3 - (7*a
- 12)*y**3}$
% clear the rules previously set.
setrules {};
{}$
% Example 11 : The standard basis of another example.
% Comparing different approaches.
vars:={x,y}$
setring(vars,localorder vars,lex);
{{x,y},{{-1,-1}},lex,{1,1}}$
ff:=x^5+y^11+(x+x^3)*y^9;
ff := x**5 + x**3*y**9 + x*y**9 + y**11$
setideal(p1,mat2list matjac({ff},vars));
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8}$
gbasis p1;
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8,
73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2*y
**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$
gbtestversion 2$
setideal(p2,p1);
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8}$
gbasis p2;
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8,
73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2*y
**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$
gbtestversion 3$
setideal(p3,p1);
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8}$
gbasis p3;
{5*x**4 + y**9 + 3*x**2*y**9,
9*x*y**8 + 11*y**10 + 9*x**3*y**8,
73205*y**16 + 6561*y**17 - 32805*x**10*y**8 + 294030*x**6*y**12 - 292820*x**2*y
**16 + 120285*x**9*y**10 - 239580*x**5*y**14 + 19683*x**2*y**17}$
gbtestversion 1$
modequalp(p1,p2);
yes$
modequalp(p1,p3);
yes$
dim p1;
0$
degree p1;
40$
% Example 12 : A local intersection wrt. a non inflimited term order.
setring({x,y,z},{},revlex);
{{x,y,z},{},revlex,{1,1,1}}$
m1:=matintersect({x-y^2,y-x^2},{x-z^2,z-x^2},{y-z^2,z-y^2});
m1 := {y*z - x**3*y*z - x*y*z**2 + x**4*y*z**2 - y**2*z**2 + x**3*y**2*z**2 + x*
y**2*z**3 - x**4*y**2*z**3,
x*z - x**2*y*z - x**2*z**2 - x*y*z**2 + x**3*y*z**2 + x**2*y**2*z**2 + x**2*y*z
**3 - x**3*y**2*z**3,
x*y - x**2*y**2 - x**2*y*z - x*y**2*z + x**3*y**2*z + x**2*y**3*z + x**2*y**2*z
**2 - x**3*y**3*z**2}$
% Delete polynomial units post factum :
deleteunits ws;
{y*z,x*z,x*y}$
% Detecting polynomial units early :
on detectunits;
m1:=matintersect({x-y^2,y-x^2},{x-z^2,z-x^2},{y-z^2,z-y^2});
m1 := {y*z,x*z,x*y}$
off detectunits;
comment
####################################
### ###
### More Advanced Computations ###
### ###
####################################
end comment;
% Return to a noetherian term order:
vars:={x,y,z}$
setring(vars,degreeorder vars,revlex);
{{x,y,z},{{1,1,1}},revlex,{1,1,1}}$
on noetherian;
% Example 13 : Use of "mod".
% Polynomials modulo ideals :
setideal(m,{2x^2+y+5,3y^2+z+7,7z^2+x+1});
{2*x**2 + y + 5,3*y**2 + z + 7,7*z**2 + x + 1}$
x^2*y^2*z^2 mod m;
( - x*y*z - 7*x*y - 5*x*z - 35*x - y*z - 7*y - 5*z - 35)/42$
% Lists of polynomials modulo ideals :
{x^3,y^3,z^3} mod gbasis m;
{(x*( - y - 5))/2,(y*( - z - 7))/3,( - z*(x + 1))/7}$
% Matrices modulo modules :
mm:=mat((x^4,y^4,z^4));
mm := mat((x**4,y**4,z**4))$
mm1:=tp<< ideal2mat m>>;
mm1 := mat((2*x**2 + y + 5,3*y**2 + z + 7,x + 7*z**2 + 1))$
mm mod mm1;
mat(((y**2 + 10*y + 25)/4,( - 6*x**2*y**2 - 2*x**2*z - 14*x**2 + 4*y**4 + 3*y**3
+ 15*y**2 + y*z + 7*y + 5*z + 35)/4,( - 2*x**3 - 14*x**2*z**2 - 2*x**2 + x*y +
5*x + 7*y*z**2 + y + 4*z**4 + 35*z**2 + 5)/4))$
% Example 14 : Powersums through elementary symmetric functions.
vars:={a,b,c,d,e1,e2,e3,e4}$
setring(vars,{},lex)$
m:=interreduce {a+b+c+d-e1,
a*b+a*c+a*d+b*c+b*d+c*d-e2,
a*b*c+a*b*d+a*c*d+b*c*d-e3,
a*b*c*d-e4};
m := {d**4 - d**3*e1 + d**2*e2 - d*e3 + e4,
a + b + c + d - e1,
c**3 + c**2*d - c**2*e1 + c*d**2 - c*d*e1 + c*e2 + d**3 - d**2*e1 + d*e2 - e3,
b**2 + b*c + b*d - b*e1 + c**2 + c*d - c*e1 + d**2 - d*e1 + e2}$
for n:=1:5 collect a^n+b^n+c^n+d^n mod m;
{e1,
e1**2 - 2*e2,
e1**3 - 3*e1*e2 + 3*e3,
e1**4 - 4*e1**2*e2 + 4*e1*e3 + 2*e2**2 - 4*e4,
e1**5 - 5*e1**3*e2 + 5*e1**2*e3 + 5*e1*e2**2 - 5*e1*e4 - 5*e2*e3}$
% Example 15 : The setrules mechanism.
setring({x,y,z},{},lex)$
setrules {aa^3=>aa+1};
{aa**3 => aa + 1}$
setideal(m,{x^2+y+z-aa,x+y^2+z-aa,x+y+z^2-aa});
{x**2 + y + z - aa,x + y**2 + z - aa,x + y + z**2 - aa}$
gbasis m;
{y**2 - y - z**2 + z,
x + y + z**2 - aa,
2*y*z**2 - (2*aa - 2)*y + z**4 - (2*aa - 1)*z**2 + (aa**2 - aa),
z**6 - (3*aa + 1)*z**4 + 4*z**3 + (3*aa**2 - 2*aa - 2)*z**2 - (4*aa - 4)*z + (3*
aa**2 - 3*aa - 1)}$
% Clear the rules previously set.
setrules {};
{}$
% Example 16 : The same example with advanced coefficient domains.
load_package arnum;
defpoly aa^3-aa-1;
setideal(m,{x^2+y+z-aa,x+y^2+z-aa,x+y+z^2-aa});
{x**2 + y + z - aa,x + y**2 + z - aa,x + y + z**2 - aa}$
gbasis m;
{y**2 - y - z**2 + z,
x + y + z**2 - aa,
y*z**2 - (aa - 1)*y + 1/2*z**4 - (aa - 1/2)*z**2 + (1/2*aa**2 - 1/2*aa),
z**6 - (3*aa + 1)*z**4 + 4*z**3 + (3*aa**2 - 2*aa - 2)*z**2 - (4*aa - 4)*z + (3*
aa**2 - 3*aa - 1)}$
% The following needs some more time since factorization of
% arnum's is not so easy :
groebfactor m;
{{y - (aa**2 - aa - 1),
z - (aa**2 - aa - 1),
x + (aa**2 - aa - 2)},
{y + (aa**2 - aa - 1),
z + (aa**2 - aa - 1),
x - (aa**2 - aa)},
{y - z,x - z,z**2 + 2*z - aa},
{z - (aa**2 - aa),
y + (aa**2 - aa - 1),
x + (aa**2 - aa - 1)},
{z - (aa**2 - aa - 1),
y + (aa**2 - aa - 2),
x - (aa**2 - aa - 1)},
{z + (aa**2 - aa - 2),
y - (aa**2 - aa - 1),
x - (aa**2 - aa - 1)},
{z + (aa**2 - aa - 1),
y - (aa**2 - aa),
x + (aa**2 - aa - 1)}}$
off arnum;
off rational;
comment
####################################
### ###
### Using Advanced Scripts in ###
### a Complex Example ###
### ###
####################################
end comment;
% Example 17 : The square of the 2-minors of a symmetric 3x3-matrix.
vars:=for i:=1:6 collect mkid(x,i);
vars := {x1,
x2,
x3,
x4,
x5,
x6}$
setring(vars,degreeorder vars,revlex);
{{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$
% Generating the ideal :
mm:=mat((x1,x2,x3),(x2,x4,x5),(x3,x5,x6));
mm := mat((x1,x2,x3),(x2,x4,x5),(x3,x5,x6))$
m:=ideal_of_minors(mm,2);
m := { - x1*x4 + x2**2,
- x1*x5 + x2*x3,
- x1*x6 + x3**2,
- x2*x5 + x3*x4,
- x2*x6 + x3*x5,
- x4*x6 + x5**2}$
setideal(n,idealpower(m,2));
{x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2,
x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2,
x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2,
x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2,
x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2,
x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5,
x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6,
x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5,
x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6,
x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6,
x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2,
x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6,
x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6,
x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6,
x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2,
x3**2*x4**2 - 2*x2*x3*x4*x5 + x1*x4*x5**2 + x2**2*x4*x6 - x1*x4**2*x6,
x2*x3**2*x4 - 2*x1*x3*x4*x5 + x1*x2*x5**2 - x2**3*x6 + x1*x2*x4*x6,
x3**3*x4 - x1*x3*x5**2 - x2**2*x3*x6 - x1*x3*x4*x6 + 2*x1*x2*x5*x6,
x3**2*x4*x5 - x1*x5**3 - 2*x2*x3*x4*x6 + x2**2*x5*x6 + x1*x4*x5*x6,
x3**2*x4*x6 - 2*x2*x3*x5*x6 + x1*x5**2*x6 + x2**2*x6**2 - x1*x4*x6**2,
x1*x3**2*x4 - 2*x1*x2*x3*x5 + x1**2*x5**2 + x1*x2**2*x6 - x1**2*x4*x6}$
% The ideal itself :
gbasis n;
{x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2,
x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2,
x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2,
x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2,
x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2,
x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5,
x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6,
x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5,
x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6,
x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6,
x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2,
x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6,
x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6,
x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6,
x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2,
x3**2*x4**2 - 2*x2*x3*x4*x5 + x1*x4*x5**2 + x2**2*x4*x6 - x1*x4**2*x6,
x2*x3**2*x4 - 2*x1*x3*x4*x5 + x1*x2*x5**2 - x2**3*x6 + x1*x2*x4*x6,
x3**3*x4 - x1*x3*x5**2 - x2**2*x3*x6 - x1*x3*x4*x6 + 2*x1*x2*x5*x6,
x3**2*x4*x5 - x1*x5**3 - 2*x2*x3*x4*x6 + x2**2*x5*x6 + x1*x4*x5*x6,
x3**2*x4*x6 - 2*x2*x3*x5*x6 + x1*x5**2*x6 + x2**2*x6**2 - x1*x4*x6**2,
x1*x3**2*x4 - 2*x1*x2*x3*x5 + x1**2*x5**2 + x1*x2**2*x6 - x1**2*x4*x6}$
length n;
21$
dim n;
3$
degree n;
16$
% Its radical.
radical n;
{ - x1*x5 + x2*x3,
- x2*x5 + x3*x4,
- x2*x6 + x3*x5,
- x1*x4 + x2**2,
- x1*x6 + x3**2,
- x4*x6 + x5**2}$
% Its unmixed radical.
unmixedradical n;
{ - x1*x5 + x2*x3,
x2*x5 - x3*x4,
- x2*x6 + x3*x5,
- x1*x4 + x2**2,
- x1*x6 + x3**2,
- x4*x6 + x5**2}$
% Its equidimensional hull :
n1:=eqhull n;
n1 := {x1**2*x4**2 - 2*x1*x2**2*x4 + x2**4,
x1**2*x6**2 - 2*x1*x3**2*x6 + x3**4,
x4**2*x6**2 - 2*x4*x5**2*x6 + x5**4,
x1**2*x5**2 - 2*x1*x2*x3*x5 + x2**2*x3**2,
x2**2*x6**2 - 2*x2*x3*x5*x6 + x3**2*x5**2,
x1**2*x4*x5 - x1*x2**2*x5 - x1*x2*x3*x4 + x2**3*x3,
x1**2*x5*x6 - x1*x2*x3*x6 - x1*x3**2*x5 + x2*x3**3,
x1*x2*x4*x5 - x1*x3*x4**2 - x2**3*x5 + x2**2*x3*x4,
x1*x2*x4*x6 - x1*x3*x4*x5 - x2**3*x6 + x2**2*x3*x5,
x1*x2*x5*x6 - x1*x3*x5**2 - x2**2*x3*x6 + x2*x3**2*x5,
x1*x2*x6**2 - x1*x3*x5*x6 - x2*x3**2*x6 + x3**3*x5,
x1*x4**2*x6 - x1*x4*x5**2 - x2**2*x4*x6 + x2**2*x5**2,
x1*x4*x5*x6 - x1*x5**3 - x2*x3*x4*x6 + x2*x3*x5**2,
x2*x4*x5*x6 - x2*x5**3 - x3*x4**2*x6 + x3*x4*x5**2,
x2*x4*x6**2 - x2*x5**2*x6 - x3*x4*x5*x6 + x3*x5**3,
- x1*x4*x6 + x1*x5**2 + x2**2*x6 - 2*x2*x3*x5 + x3**2*x4}$
length n1;
16$
setideal(n1,n1)$
submodulep(n,n1);
yes$
submodulep(n1,n);
no$
% Hence there is an embedded component. Let's find it making
% an excursion to symbolic mode. Of course, this can be done
% also algebraically.
symbolic;
nil
n:=get('n,'basis);
(dpmat 21 0 ((1 ((((0 0 4) 4) . 1) (((0 1 2 0 1) 4) . -2) (((0 2 0 0 2) 4) . 1))
3 0 nil) (2 ((((0 0 0 4) 4) . 1) (((0 1 0 2 0 0 1) 4) . -2) (((0 2 0 0 0 0 2) 4)
. 1)) 3 0 nil) (3 ((((0 0 0 0 0 4) 4) . 1) (((0 0 0 0 1 2 1) 4) . -2) (((0 0 0 0
2 0 2) 4) . 1)) 3 0 nil) (4 ((((0 0 2 2) 4) . 1) (((0 1 1 1 0 1) 4) . -2) (((0 2
0 0 0 2) 4) . 1)) 3 0 nil) (5 ((((0 0 0 2 0 2) 4) . 1) (((0 0 1 1 0 1 1) 4) . -2
) (((0 0 2 0 0 0 2) 4) . 1)) 3 0 nil) (6 ((((0 0 3 1) 4) . 1) (((0 1 1 1 1) 4) .
-1) (((0 1 2 0 0 1) 4) . -1) (((0 2 0 0 1 1) 4) . 1)) 4 0 nil) (7 ((((0 0 1 3) 4
) . 1) (((0 1 0 2 0 1) 4) . -1) (((0 1 1 1 0 0 1) 4) . -1) (((0 2 0 0 0 1 1) 4)
. 1)) 4 0 nil) (8 ((((0 0 2 1 1) 4) . 1) (((0 1 0 1 2) 4) . -1) (((0 0 3 0 0 1)
4) . -1) (((0 1 1 0 1 1) 4) . 1)) 4 0 nil) (9 ((((0 0 2 1 0 1) 4) . 1) (((0 1 0
1 1 1) 4) . -1) (((0 0 3 0 0 0 1) 4) . -1) (((0 1 1 0 1 0 1) 4) . 1)) 4 0 nil) (
10 ((((0 0 1 2 0 1) 4) . 1) (((0 1 0 1 0 2) 4) . -1) (((0 0 2 1 0 0 1) 4) . -1)
(((0 1 1 0 0 1 1) 4) . 1)) 4 0 nil) (11 ((((0 0 0 3 0 1) 4) . 1) (((0 0 1 2 0 0
1) 4) . -1) (((0 1 0 1 0 1 1) 4) . -1) (((0 1 1 0 0 0 2) 4) . 1)) 4 0 nil) (12 (
(((0 0 2 0 0 2) 4) . 1) (((0 1 0 0 1 2) 4) . -1) (((0 0 2 0 1 0 1) 4) . -1) (((0
1 0 0 2 0 1) 4) . 1)) 4 0 nil) (13 ((((0 0 1 1 0 2) 4) . 1) (((0 1 0 0 0 3) 4) .
-1) (((0 0 1 1 1 0 1) 4) . -1) (((0 1 0 0 1 1 1) 4) . 1)) 4 0 nil) (14 ((((0 0 0
1 1 2) 4) . 1) (((0 0 1 0 0 3) 4) . -1) (((0 0 0 1 2 0 1) 4) . -1) (((0 0 1 0 1
1 1) 4) . 1)) 4 0 nil) (15 ((((0 0 0 1 0 3) 4) . 1) (((0 0 0 1 1 1 1) 4) . -1) (
((0 0 1 0 0 2 1) 4) . -1) (((0 0 1 0 1 0 2) 4) . 1)) 4 0 nil) (16 ((((0 0 0 2 2)
4) . 1) (((0 0 1 1 1 1) 4) . -2) (((0 1 0 0 1 2) 4) . 1) (((0 0 2 0 1 0 1) 4) .
1) (((0 1 0 0 2 0 1) 4) . -1)) 5 0 nil) (17 ((((0 0 1 2 1) 4) . 1) (((0 1 0 1 1
1) 4) . -2) (((0 1 1 0 0 2) 4) . 1) (((0 0 3 0 0 0 1) 4) . -1) (((0 1 1 0 1 0 1)
4) . 1)) 5 0 nil) (18 ((((0 0 0 3 1) 4) . 1) (((0 1 0 1 0 2) 4) . -1) (((0 0 2 1
0 0 1) 4) . -1) (((0 1 0 1 1 0 1) 4) . -1) (((0 1 1 0 0 1 1) 4) . 2)) 5 0 nil) (
19 ((((0 0 0 2 1 1) 4) . 1) (((0 1 0 0 0 3) 4) . -1) (((0 0 1 1 1 0 1) 4) . -2)
(((0 0 2 0 0 1 1) 4) . 1) (((0 1 0 0 1 1 1) 4) . 1)) 5 0 nil) (20 ((((0 0 0 2 1
0 1) 4) . 1) (((0 0 1 1 0 1 1) 4) . -2) (((0 1 0 0 0 2 1) 4) . 1) (((0 0 2 0 0 0
2) 4) . 1) (((0 1 0 0 1 0 2) 4) . -1)) 5 0 nil) (21 ((((0 1 0 2 1) 4) . 1) (((0
1 1 1 0 1) 4) . -2) (((0 2 0 0 0 2) 4) . 1) (((0 1 2 0 0 0 1) 4) . 1) (((0 2 0 0
1 0 1) 4) . -1)) 5 0 nil)) nil nil)
% This needs even more time than the eqhull, of course.
u:=primarydecomposition!* n;
(((dpmat 16 0 ((1 ((((0 0 2 1 0 1) 4) . 1) (((0 1 0 1 1 1) 4) . -1) (((0 0 3 0 0
0 1) 4) . -1) (((0 1 1 0 1 0 1) 4) . 1)) 4 0 nil) (2 ((((0 0 1 2 0 1) 4) . 1) ((
(0 1 0 1 0 2) 4) . -1) (((0 0 2 1 0 0 1) 4) . -1) (((0 1 1 0 0 1 1) 4) . 1)) 4 0
nil) (3 ((((0 0 1 1 0 2) 4) . 1) (((0 1 0 0 0 3) 4) . -1) (((0 0 1 1 1 0 1) 4) .
-1) (((0 1 0 0 1 1 1) 4) . 1)) 4 0 nil) (4 ((((0 0 4) 4) . 1) (((0 1 2 0 1) 4) .
-2) (((0 2 0 0 2) 4) . 1)) 3 0 nil) (5 ((((0 0 3 1) 4) . 1) (((0 1 1 1 1) 4) .
-1) (((0 1 2 0 0 1) 4) . -1) (((0 2 0 0 1 1) 4) . 1)) 4 0 nil) (6 ((((0 0 2 2) 4
) . 1) (((0 1 1 1 0 1) 4) . -2) (((0 2 0 0 0 2) 4) . 1)) 3 0 nil) (7 ((((0 0 2 1
1) 4) . 1) (((0 1 0 1 2) 4) . -1) (((0 0 3 0 0 1) 4) . -1) (((0 1 1 0 1 1) 4) .
1)) 4 0 nil) (8 ((((0 0 2 0 0 2) 4) . 1) (((0 1 0 0 1 2) 4) . -1) (((0 0 2 0 1 0
1) 4) . -1) (((0 1 0 0 2 0 1) 4) . 1)) 4 0 nil) (9 ((((0 0 0 4) 4) . 1) (((0 1 0
2 0 0 1) 4) . -2) (((0 2 0 0 0 0 2) 4) . 1)) 3 0 nil) (10 ((((0 0 0 0 0 4) 4) .
1) (((0 0 0 0 1 2 1) 4) . -2) (((0 0 0 0 2 0 2) 4) . 1)) 3 0 nil) (11 ((((0 0 0
2 0 2) 4) . 1) (((0 0 1 1 0 1 1) 4) . -2) (((0 0 2 0 0 0 2) 4) . 1)) 3 0 nil) (
12 ((((0 0 1 3) 4) . 1) (((0 1 0 2 0 1) 4) . -1) (((0 1 1 1 0 0 1) 4) . -1) (((0
2 0 0 0 1 1) 4) . 1)) 4 0 nil) (13 ((((0 0 0 3 0 1) 4) . 1) (((0 0 1 2 0 0 1) 4)
. -1) (((0 1 0 1 0 1 1) 4) . -1) (((0 1 1 0 0 0 2) 4) . 1)) 4 0 nil) (14 ((((0 0
0 1 1 2) 4) . 1) (((0 0 1 0 0 3) 4) . -1) (((0 0 0 1 2 0 1) 4) . -1) (((0 0 1 0
1 1 1) 4) . 1)) 4 0 nil) (15 ((((0 0 0 1 0 3) 4) . 1) (((0 0 0 1 1 1 1) 4) . -1)
(((0 0 1 0 0 2 1) 4) . -1) (((0 0 1 0 1 0 2) 4) . 1)) 4 0 nil) (16 ((((0 0 0 2 1
) 3) . 1) (((0 0 1 1 0 1) 3) . -2) (((0 1 0 0 0 2) 3) . 1) (((0 0 2 0 0 0 1) 3)
. 1) (((0 1 0 0 1 0 1) 3) . -1)) 5 0 nil)) nil t) (dpmat 6 0 ((1 ((((0 0 0 1 0 1
) 2) . 1) (((0 0 1 0 0 0 1) 2) . -1)) 2 0 nil) (2 ((((0 0 0 0 0 2) 2) . 1) (((0
0 0 0 1 0 1) 2) . -1)) 2 0 nil) (3 ((((0 0 0 1 1) 2) . 1) (((0 0 1 0 0 1) 2) .
-1)) 2 0 nil) (4 ((((0 0 0 2) 2) . 1) (((0 1 0 0 0 0 1) 2) . -1)) 2 0 nil) (5 ((
((0 0 1 1) 2) . 1) (((0 1 0 0 0 1) 2) . -1)) 2 0 nil) (6 ((((0 0 2) 2) . 1) (((0
1 0 0 1) 2) . -1)) 2 0 nil)) nil t)) ((dpmat 18 0 ((1 ((((0 0 1 0 0 3) 4) . 1))
1 0 nil) (2 ((((0 0 0 0 0 0 1) 1) . 1)) 1 0 nil) (3 ((((0 0 0 4) 4) . 1)) 1 0
nil) (4 ((((0 0 0 0 0 4) 4) . 1)) 1 0 nil) (5 ((((0 0 0 2 0 2) 4) . 1)) 1 0 nil)
(6 ((((0 0 0 3 0 1) 4) . 1)) 1 0 nil) (7 ((((0 0 0 1 0 3) 4) . 1)) 1 0 nil) (8 (
(((0 0 0 0 1) 1) . 1)) 1 0 nil) (9 ((((0 0 3 0 0 1) 4) . 1)) 1 0 nil) (10 ((((0
0 2 1 0 1) 4) . 1)) 1 0 nil) (11 ((((0 0 1 2 0 1) 4) . 1)) 1 0 nil) (12 ((((0 0
2 0 0 2) 4) . 1)) 1 0 nil) (13 ((((0 0 1 1 0 2) 4) . 1)) 1 0 nil) (14 ((((0 0 4)
4) . 1)) 1 0 nil) (15 ((((0 0 2 2) 4) . 1)) 1 0 nil) (16 ((((0 1) 1) . 1)) 1 0
nil) (17 ((((0 0 1 3) 4) . 1)) 1 0 nil) (18 ((((0 0 3 1) 4) . 1)) 1 0 nil)) nil
t) (dpmat 6 0 ((1 ((((0 0 0 0 0 0 1) 1) . 1)) 1 0 nil) (2 ((((0 0 0 0 1) 1) . 1)
) 1 0 nil) (3 ((((0 1) 1) . 1)) 1 0 nil) (4 ((((0 0 1) 1) . 1)) 1 0 nil) (5 ((((
0 0 0 1) 1) . 1)) 1 0 nil) (6 ((((0 0 0 0 0 1) 1) . 1)) 1 0 nil)) nil t)))
for each x in u collect easydim!* cadr x;
(3 0)
for each x in u collect degree!* car x;
(16 20)
% Hence the embedded component is a trivial one. Let's divide
% it out by a stable ideal quotient calculation :
algebraic;
setideal(n2,matstabquot(n,vars));
{x2**4 - 2*x1*x2**2*x4 + x1**2*x4**2,
x3**4 - 2*x1*x3**2*x6 + x1**2*x6**2,
x5**4 - 2*x4*x5**2*x6 + x4**2*x6**2,
x2**2*x3**2 - 2*x1*x2*x3*x5 + x1**2*x5**2,
x3**2*x5**2 - 2*x2*x3*x5*x6 + x2**2*x6**2,
x2**3*x3 - x1*x2*x3*x4 - x1*x2**2*x5 + x1**2*x4*x5,
x2*x3**3 - x1*x3**2*x5 - x1*x2*x3*x6 + x1**2*x5*x6,
x2**2*x3*x4 - x1*x3*x4**2 - x2**3*x5 + x1*x2*x4*x5,
x2**2*x3*x5 - x1*x3*x4*x5 - x2**3*x6 + x1*x2*x4*x6,
x2*x3**2*x5 - x1*x3*x5**2 - x2**2*x3*x6 + x1*x2*x5*x6,
x3**3*x5 - x2*x3**2*x6 - x1*x3*x5*x6 + x1*x2*x6**2,
x2**2*x5**2 - x1*x4*x5**2 - x2**2*x4*x6 + x1*x4**2*x6,
x2*x3*x5**2 - x1*x5**3 - x2*x3*x4*x6 + x1*x4*x5*x6,
x3*x4*x5**2 - x2*x5**3 - x3*x4**2*x6 + x2*x4*x5*x6,
x3*x5**3 - x3*x4*x5*x6 - x2*x5**2*x6 + x2*x4*x6**2,
x3**2*x4 - 2*x2*x3*x5 + x1*x5**2 + x2**2*x6 - x1*x4*x6}$
modequalp(n1,n2);
yes$
comment
########################################
### ###
### Test Examples for New Features ###
### ###
########################################
end comment;
% ==> Testing the different zerodimensional solver
vars:={x,y,z}$
setring(vars,degreeorder vars,revlex);
{{x,y,z},{{1,1,1}},revlex,{1,1,1}}$
setideal(m,{x^3+y+z-3,y^3+x+z-3,z^3+x+y-3});
{x**3 + y + z - 3,y**3 + x + z - 3,z**3 + x + y - 3}$
zerosolve1 m;
{{x + y + z**3 - 3,
y + z**3 + z - 3,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x + y + z**3 - 3,
2*y + z**3 - 3,
z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5},
{x + y + z,
y**2 + y*z + z**2 - 1,
z**3 - z - 3},
{x + z**3 + z - 3,
y - z,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x - z,y - z,z**2 + z + 3},
{x - 1,y - 1,z - 1}}$
zerosolve2 m;
{{x + y + z**3 - 3,
y + z**3 + z - 3,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x + y + z**3 - 3,
2*y + z**3 - 3,
z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5},
{x + y + z,
y**2 + y*z + z**2 - 1,
z**3 - z - 3},
{x + z**3 + z - 3,
y - z,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x - z,y - z,z**2 + z + 3},
{x - 1,y - 1,z - 1}}$
setring(vars,{},lex)$
setideal(m,m)$
m1:=gbasis m$
zerosolve m1;
{{x - 1,y - 1,z - 1},
{x - z,y - z,z**2 + z + 3},
{x + y + z,
y**2 + y*z + z**2 - 1,
z**3 - z - 3},
{2*x + z**3 - 3,
2*y + z**3 - 3,
z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5},
{x + z**3 + z - 3,
y - z,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x - z,
y + z**3 + z - 3,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$
zerosolve1 m1;
{{x - 1,y - 1,z - 1},
{x - z,y - z,z**2 + z + 3},
{x + y + z,
y**2 + y*z + z**2 - 1,
z**3 - z - 3},
{x - y,
2*y + z**3 - 3,
z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5},
{x + z**3 + z - 3,
y - z,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x - z,
y + z**3 + z - 3,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$
zerosolve2 m1;
{{x - 1,y - 1,z - 1},
{x - z,y - z,z**2 + z + 3},
{x + y + z,
y**2 + y*z + z**2 - 1,
z**3 - z - 3},
{x - y,
2*y + z**3 - 3,
z**6 - 2*z**4 - 6*z**3 + 4*z**2 + 6*z + 5},
{x + z**3 + z - 3,
y - z,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8},
{x - z,
y + z**3 + z - 3,
z**6 + z**4 - 6*z**3 + z**2 - 3*z + 8}}$
% ==> Testing groebfactor, extendedgroebfactor, extendedgroebfactor1
% Gerdt et al. : Seventh order KdV type equation.
A1:=-2*L1**2+L1*L2+2*L1*L3-L2**2-7*L5+21*L6$
A2:=7*L7-2*L1*L4+3/7*L1**3$
B1:=L1*(5*L1-3*L2+L3)$
B2:=L1*(2*L6-4*L4)$
B3:=L1*L7/2$
P1:=L1*(L4-L5/2+L6)$
P2:=(2/7*L1**2-L4)*(-10*L1+5*L2-L3)$
P3:=(2/7*L1**2-L4)*(3*L4-L5+L6)$
P4:=A1*(-3*L1+2*L2)+21*A2$
P5:=A1*(2*L4-2*L5)+A2*(-45*L1+15*L2-3*L3)$
P6:=2*A1*L7+A2*(12*L4-3*L5+2*L6)$
P7:=B1*(2*L2-L1)+7*B2$
P8:=B1*L3+7*B2$
P9:=B1*(-2*L4-2*L5)+B2*(2*L2-8*L1)+84*B3$
P10:=B1*(8/3*L5+6*L6)+B2*(11*L1-17/3*L2+5/3*L3)-168*B3$
P11:=15*B1*L7+B2*(5*L4-2*L5)+B3*(-120*L1+30*L2-6*L3)$
P12:=-3*B1*L7+B2*(-L4/2+L5/4-L6/2)+B3*(24*L1-6*L2)$
P13:=3*B2*L7+B3*(40*L4-8*L5+4*L6)$
polys:={P1,P2,P3,P4,P5,P6,P7,P8,P9,P10,P11,P12,P13};
polys := {(l1*(2*l4 - l5 + 2*l6))/2,
( - 20*l1**3 + 10*l1**2*l2 - 2*l1**2*l3 + 70*l1*l4 - 35*l2*l4 + 7*l3*l4)/7,
(6*l1**2*l4 - 2*l1**2*l5 + 2*l1**2*l6 - 21*l4**2 + 7*l4*l5 - 7*l4*l6)/7,
15*l1**3 - 7*l1**2*l2 - 6*l1**2*l3 + 5*l1*l2**2 + 4*l1*l2*l3 - 42*l1*l4 + 21*l1*
l5 - 63*l1*l6 - 2*l2**3 - 14*l2*l5 + 42*l2*l6 + 147*l7,
( - 135*l1**4 + 45*l1**3*l2 - 9*l1**3*l3 + 602*l1**2*l4 + 28*l1**2*l5 - 196*l1*
l2*l4 - 14*l1*l2*l5 + 70*l1*l3*l4 - 28*l1*l3*l5 - 2205*l1*l7 - 14*l2**2*l4 + 14*
l2**2*l5 + 735*l2*l7 - 147*l3*l7 - 98*l4*l5 + 294*l4*l6 + 98*l5**2 - 294*l5*l6)/
7,
(36*l1**3*l4 - 9*l1**3*l5 + 6*l1**3*l6 - 28*l1**2*l7 + 14*l1*l2*l7 + 28*l1*l3*l7
- 168*l1*l4**2 + 42*l1*l4*l5 - 28*l1*l4*l6 - 14*l2**2*l7 + 588*l4*l7 - 245*l5*
l7 + 392*l6*l7)/7,
l1*( - 5*l1**2 + 13*l1*l2 - l1*l3 - 6*l2**2 + 2*l2*l3 - 28*l4 + 14*l6),
l1*(5*l1*l3 - 3*l2*l3 + l3**2 - 28*l4 + 14*l6),
2*l1*(11*l1*l4 - 5*l1*l5 - 8*l1*l6 - l2*l4 + 3*l2*l5 + 2*l2*l6 - l3*l4 - l3*l5 +
21*l7),
(4*l1*( - 33*l1*l4 + 10*l1*l5 + 39*l1*l6 + 17*l2*l4 - 6*l2*l5 - 22*l2*l6 - 5*l3*
l4 + 2*l3*l5 + 7*l3*l6 - 63*l7))/3,
l1*(15*l1*l7 - 30*l2*l7 + 12*l3*l7 - 20*l4**2 + 8*l4*l5 + 10*l4*l6 - 4*l5*l6),
(l1*( - 6*l1*l7 + 12*l2*l7 - 6*l3*l7 + 4*l4**2 - 2*l4*l5 + 2*l4*l6 + l5*l6 - 2*
l6**2))/2,
4*l1*l7*(2*l4 - l5 + 2*l6)}$
vars:={L7,L6,L5,L4,L3,L2,L1};
vars := {l7,
l6,
l5,
l4,
l3,
l2,
l1}$
clear a1,a2,b1,b2,b3$
off lexefgb;
setring(vars,{},lex);
{{l7,l6,l5,l4,l3,l2,l1},{},lex,{1,1,1,1,1,1,1}}$
% The factorized Groebner algorithm.
groebfactor polys;
{{l1,l4,l7,21*l6 - 7*l5 - l2**2},
{l1,
l4,
7*l5 - l3*l2 + 5*l2**2,
56*l6 - 5*l3*l2 + 23*l2**2,
588*l7 + 7*l3*l2**2 - 37*l2**3},
{l1,
l7,
l3 - 5*l2,
14*l6 - 21*l4 - l2**2,
14*l5 - 63*l4 - l2**2},
{l1,l4,l2,l5,l7},
{7*l4 - 2*l1**2,
l2 - 2*l1,
l3 - 3*l1,
147*l7 - 4*l1**3,
7*l5 - 6*l1**2,
7*l6 - l1**2},
{7*l4 - 2*l1**2,
2*l2 - 7*l1,
l3 - 6*l1,
147*l7 - 4*l1**3,
7*l5 - 9*l1**2,
14*l6 - 5*l1**2},
{l1,
l3 - 5*l2,
63*l4 + 2*l2**2,
63*l5 + 2*l2**2,
63*l6 - 4*l2**2,
1323*l7 + 10*l2**3},
{l1,l2,l3,l7,l5 - l4,l6 + 2*l4},
{l2 - 3*l1,
l3 - 5*l1,
14*l4 - 5*l1**2,
98*l7 - 5*l1**3,
7*l5 - 10*l1**2,
14*l6 - 5*l1**2}}$
% The extended Groebner factorizer, producing triangular sets.
extendedgroebfactor polys;
{{{98*l7 - 5*l1**3,
14*l6 - 5*l1**2,
7*l5 - 10*l1**2,
14*l4 - 5*l1**2,
l3 - 5*l1,
l2 - 3*l1},
{},
{l1}},
{{l7,l6 + 2*l4,l5 - l4,l3,l2,l1},{},{l4}},
{{1323*l7 + 10*l2**3,
63*l6 - 4*l2**2,
63*l5 + 2*l2**2,
63*l4 + 2*l2**2,
l3 - 5*l2,
l1},
{},
{l2}},
{{147*l7 - 4*l1**3,
14*l6 - 5*l1**2,
7*l5 - 9*l1**2,
7*l4 - 2*l1**2,
l3 - 6*l1,
2*l2 - 7*l1},
{},
{l1}},
{{147*l7 - 4*l1**3,
7*l6 - l1**2,
7*l5 - 6*l1**2,
7*l4 - 2*l1**2,
l3 - 3*l1,
l2 - 2*l1},
{},
{l1}},
{{l7,l5,l4,l2,l1},{},{l6,l3}},
{{l7,
14*l6 - (l2**2 + 21*l4),
14*l5 - (l2**2 + 63*l4),
l3 - 5*l2,
l1},
{},
{l4,l2}},
{{588*l7 - (37*l2**3 - 7*l2**2*l3),
56*l6 + (23*l2**2 - 5*l2*l3),
7*l5 + (5*l2**2 - l2*l3),
l4,
l1},
{},
{l3,l2}},
{{l7,21*l6 - (l2**2 + 7*l5),l4,l1},{},{l5,l3,l2}}}$
% The extended Groebner factorizer with subproblem removal check.
extendedgroebfactor1 polys;
{{{l7,21*l6 - (l2**2 + 7*l5),l4,l1},{},{l5,l3,l2}},
{{588*l7 - (37*l2**3 - 7*l2**2*l3),
56*l6 + (23*l2**2 - 5*l2*l3),
7*l5 + (5*l2**2 - l2*l3),
l4,
l1},
{},
{l3,l2}},
{{l7,
14*l6 - (l2**2 + 21*l4),
14*l5 - (l2**2 + 63*l4),
l3 - 5*l2,
l1},
{},
{l4,l2}},
{{l7,l5,l4,l2,l1},{},{l6,l3}},
{{147*l7 - 4*l1**3,
7*l6 - l1**2,
7*l5 - 6*l1**2,
7*l4 - 2*l1**2,
l3 - 3*l1,
l2 - 2*l1},
{},
{l1}},
{{147*l7 - 4*l1**3,
14*l6 - 5*l1**2,
7*l5 - 9*l1**2,
7*l4 - 2*l1**2,
l3 - 6*l1,
2*l2 - 7*l1},
{},
{l1}},
{{1323*l7 + 10*l2**3,
63*l6 - 4*l2**2,
63*l5 + 2*l2**2,
63*l4 + 2*l2**2,
l3 - 5*l2,
l1},
{},
{l2}},
{{l7,l6 + 2*l4,l5 - l4,l3,l2,l1},{},{l4}},
{{98*l7 - 5*l1**3,
14*l6 - 5*l1**2,
7*l5 - 10*l1**2,
14*l4 - 5*l1**2,
l3 - 5*l1,
l2 - 3*l1},
{},
{l1}}}$
% Gonnet's example (ACM SIGSAM Bulletin 17 (1983), 48 - 49)
vars:={a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5};
vars := {a0,
a2,
a3,
a4,
a5,
b0,
b1,
b2,
b3,
b4,
b5,
c0,
c1,
c2,
c3,
c4,
c5}$
polys:={a4*b4,
a5*b1+b5+a4*b3+a3*b4,
a2*b2,a5*b5,
(a0+1+a4)*b2+a2*(b0+b1+b4)+c2,
(a0+1+a4)*(b0+b1+b4)+(a3+a5)*b2+a2*(b3+b5)+c0+c1+c4,
(a3+a5)*(b0+b1+b4)+(b3+b5)*(a0+1+a4)+c3+c5-1,
(a3+a5)*(b3+b5),
a5*(b3+b5)+b5*(a3+a5),
b5*(a0+1+2*a4)+a5*(b0+b1+2*b4)+a3*b4+a4*b3+c5,
a4*(b0+b1+2*b4)+a2*b5+a5*b2+(a0+1)*b4+c4,
a2*b4+a4*b2,
a4*b5+a5*b4,
2*a3*b3+a3*b5+a5*b3,
c3+b3*(a0+2+a4)+a3*(b0+2*b1+b4)+b5+a5*b1,
c1+(a0+2+a4)*b1+a2*b3+a3*b2+(b0+b4),
a2*b1+b2,
a5*b3+a3*b5,
b4+a4*b1};
polys := {a4*b4,
a3*b4 + a4*b3 + a5*b1 + b5,
a2*b2,
a5*b5,
a0*b2 + a2*b0 + a2*b1 + a2*b4 + a4*b2 + b2 + c2,
a0*b0 + a0*b1 + a0*b4 + a2*b3 + a2*b5 + a3*b2 + a4*b0 + a4*b1 + a4*b4 + a5*b2 +
b0 + b1 + b4 + c0 + c1 + c4,
a0*b3 + a0*b5 + a3*b0 + a3*b1 + a3*b4 + a4*b3 + a4*b5 + a5*b0 + a5*b1 + a5*b4 +
b3 + b5 + c3 + c5 - 1,
a3*b3 + a3*b5 + a5*b3 + a5*b5,
a3*b5 + a5*b3 + 2*a5*b5,
a0*b5 + a3*b4 + a4*b3 + 2*a4*b5 + a5*b0 + a5*b1 + 2*a5*b4 + b5 + c5,
a0*b4 + a2*b5 + a4*b0 + a4*b1 + 2*a4*b4 + a5*b2 + b4 + c4,
a2*b4 + a4*b2,
a4*b5 + a5*b4,
2*a3*b3 + a3*b5 + a5*b3,
a0*b3 + a3*b0 + 2*a3*b1 + a3*b4 + a4*b3 + a5*b1 + 2*b3 + b5 + c3,
a0*b1 + a2*b3 + a3*b2 + a4*b1 + b0 + 2*b1 + b4 + c1,
a2*b1 + b2,
a3*b5 + a5*b3,
a4*b1 + b4}$
on lexefgb;
% Switching back to the default.
setring(vars,{},lex);
{{a0,a2,a3,a4,a5,b0,b1,b2,b3,b4,b5,c0,c1,c2,c3,c4,c5},
{},
lex,
{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}}$
groebfactor polys;
{{c5,
b5,
c2,
c4,
b2,
b4,
a4,
a2,
a5,
b3,
a3*b1 + 1,
b0 - b1*c3 + 2*b1,
a0 - a3*c1 + c3,
b1*c3**2 - 2*b1*c3 + b1 - c0 + c1*c3 - 2*c1,
a3*c0 - a3*c1*c3 + 2*a3*c1 + c3**2 - 2*c3 + 1},
{c5,
c4,
b5,
b4,
a4,
b2,
a5,
a3,
b1,
b3 + 1,
a0 - c3 + 2,
b0*c3 - 2*b0 + c0,
a2 - b0 - c1,
b0**2 + b0*c1 + c2,
b0*c0 + c0*c1 - c2*c3 + 2*c2,
c0**2 - c0*c1*c3 + 2*c0*c1 + c2*c3**2 - 4*c2*c3 + 4*c2},
{c5,
b5,
c2,
c4,
b2,
b4,
a4,
a2,
a5,
a3,
b3 + 1,
b0 + b1*c3 + c1,
a0 - c3 + 2,
b1*c3**2 - 2*b1*c3 + b1 - c0 + c1*c3 - 2*c1}}$
extendedgroebfactor polys;
{{{b1*a0 + (b1 + c1),
a2,
b1*a3 + 1,
a4,
a5,
b0 + b1,
b2,
b3,
b4,
b5,
c0 + c1,
c2,
c3 - 1,
c4,
c5},
{b1,b1},
{b1,c1}},
{{a0,
a2 - b0 - c1,
a3,
a4,
a5,
b0**2 + c1*b0 + c2,
b1,
b2,
b3 + 1,
b4,
b5,
c0,
c3 - 2,
c4,
c5},
{},
{c1,c2}},
{{a0 + 1,a2,a3,a4,a5,b0 + (b1 + c1),b2,b3 + 1,b4,b5,c0 + c1,c2,c3 - 1,c4,c5},
{},
{b1,c1}},
{{a0 - (c3 - 2),
a2,
a3,
a4,
a5,
(c3**2 - 2*c3 + 1)*b0 + (c0*c3 + c1),
(c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1),
b2,
b3 + 1,
b4,
b5,
c2,
c4,
c5},
{c3**2 - 2*c3 + 1,c3**2 - 2*c3 + 1},
{c0,c1,c3}},
{{a0 - (c3 - 2),
(c3 - 2)*a2 + c0 - (c1*c3 - 2*c1),
a3,
a4,
a5,
(c3 - 2)*b0 + c0,
b1,
b2,
b3 + 1,
b4,
b5,
c0**2 - (c1*c3 - 2*c1)*c0 + (c2*c3**2 - 4*c2*c3 + 4*c2),
c4,
c5},
{c3 - 2,c3 - 2},
{c1,c2,c3}},
{{(c0 - c1*c3 + 2*c1)*a0 + (c0*c3 + c1),
a2,
(c0 - c1*c3 + 2*c1)*a3 + (c3**2 - 2*c3 + 1),
a4,
a5,
(c3**2 - 2*c3 + 1)*b0 - (c0*c3 - 2*c0 - c1*c3**2 + 4*c1*c3 - 4*c1),
(c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1),
b2,
b3,
b4,
b5,
c2,
c4,
c5},
{c0 - c1*c3 + 2*c1,
c0 - c1*c3 + 2*c1,
c3**2 - 2*c3 + 1,
c3**2 - 2*c3 + 1},
{c0,c1,c3}}}$
extendedgroebfactor1 polys;
{{{(c0 - c1*c3 + 2*c1)*a0 + (c0*c3 + c1),
a2,
(c0 - c1*c3 + 2*c1)*a3 + (c3**2 - 2*c3 + 1),
a4,
a5,
(c3**2 - 2*c3 + 1)*b0 - (c0*c3 - 2*c0 - c1*c3**2 + 4*c1*c3 - 4*c1),
(c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1),
b2,
b3,
b4,
b5,
c2,
c4,
c5},
{c0 - c1*c3 + 2*c1,
c0 - c1*c3 + 2*c1,
c3**2 - 2*c3 + 1,
c3**2 - 2*c3 + 1},
{c0,c1,c3}},
{{a0 - (c3 - 2),
(c3 - 2)*a2 + c0 - (c1*c3 - 2*c1),
a3,
a4,
a5,
(c3 - 2)*b0 + c0,
b1,
b2,
b3 + 1,
b4,
b5,
c0**2 - (c1*c3 - 2*c1)*c0 + (c2*c3**2 - 4*c2*c3 + 4*c2),
c4,
c5},
{c3 - 2,c3 - 2},
{c1,c2,c3}},
{{a0 - (c3 - 2),
a2,
a3,
a4,
a5,
(c3**2 - 2*c3 + 1)*b0 + (c0*c3 + c1),
(c3**2 - 2*c3 + 1)*b1 - (c0 - c1*c3 + 2*c1),
b2,
b3 + 1,
b4,
b5,
c2,
c4,
c5},
{c3**2 - 2*c3 + 1,c3**2 - 2*c3 + 1},
{c0,c1,c3}}}$
% Schwarz' example s5
vars:=for k:=1:5 collect mkid(x,k);
vars := {x1,
x2,
x3,
x4,
x5}$
s5:={
x1**2+x1+2*x2*x5+2*x3*x4,
2*x1*x2+x2+2*x3*x5+x4**2,
2*x1*x3+x2**2+x3+2*x4*x5,
2*x1*x4+2*x2*x3+x4+x5**2,
2*x1*x5+2*x2*x4+x3**2+x5};
s5 := {x1**2 + x1 + 2*x2*x5 + 2*x3*x4,
2*x1*x2 + x2 + 2*x3*x5 + x4**2,
2*x1*x3 + x2**2 + x3 + 2*x4*x5,
2*x1*x4 + 2*x2*x3 + x4 + x5**2,
2*x1*x5 + 2*x2*x4 + x3**2 + x5}$
setring(vars,degreeorder vars,revlex);
{{x1,x2,x3,x4,x5},{{1,1,1,1,1}},revlex,{1,1,1,1,1}}$
m:=groebfactor s5;
m := {{x1**2 + 2*x3*x4 + 2*x2*x5 + x1,
2*x1*x2 + x4**2 + 2*x3*x5 + x2,
x2**2 + 2*x1*x3 + 2*x4*x5 + x3,
2*x2*x3 + 2*x1*x4 + x5**2 + x4,
x3**2 + 2*x2*x4 + 2*x1*x5 + x5,
2*x1*x3*x4 + 2*x4**2*x5 + x3*x5**2 + x3*x4,
5*x4**3 + 30*x3*x4*x5 + 15*x2*x5**2 - 2*x5,
10*x3*x4**2 - 10*x1*x5**2 - 5*x5**2 - x4,
625*x4*x5**3 + 50*x3*x4 + 75*x2*x5 - 6,
15*x2*x4**2 + 30*x1*x4*x5 + 5*x5**3 + 15*x4*x5 + x3,
100*x1*x4*x5**2 + 25*x5**4 + 50*x4*x5**2 + x4**2 + 4*x3*x5,
1250*x1*x3*x5**2 + 625*x3*x5**2 - 75*x3*x4 - 50*x2*x5 + 8,
75*x4**2*x5**2 + 50*x3*x5**3 + x2*x4 + 4*x1*x5 + 2*x5,
150*x3*x4*x5**2 + 100*x2*x5**3 - 2*x1*x4 - 13*x5**2 - x4,
625*x2*x5**4 - 50*x1*x4*x5 - 75*x5**3 - 25*x4*x5 - x3,
1250*x3*x5**4 - 200*x2*x4*x5 - 50*x1*x5**2 - 25*x5**2 + 3*x4,
625*x5**5 + 375*x4**2*x5 + 500*x3*x5**2 + 24*x1 + 12,
10*x1*x4**2 + 20*x1*x3*x5 + 20*x4*x5**2 + 5*x4**2 + 10*x3*x5 + x2,
75*x2*x4*x5**2 + 50*x1*x5**3 + 25*x5**3 - 2*x1*x3 - 3*x4*x5 - x3,
1250*x1*x5**4 + 625*x5**4 + 100*x1*x3*x5 + 150*x4*x5**2 + 50*x3*x5 + 3*x2},
{x5,x2,x4,x3,x1},
{x5,x2,x4,x3,x1 + 1}}$
% Recompute a list of problems with listgroebfactor for another term
% order.
setring(vars,{},lex);
{{x1,x2,x3,x4,x5},{},lex,{1,1,1,1,1}}$
listgroebfactor m;
{{5*x5 - 1,
5*x4 - 1,
5*x1 + 4,
5*x2 - 1,
5*x3 - 1},
{5*x5 + 1,
5*x4 + 1,
5*x1 + 1,
5*x2 + 1,
5*x3 + 1},
{5*x1 + 2,
x2 - x5,
25*x5**2 - 5*x5 - 1,
5*x4 + 5*x5 - 1,
5*x3 + 5*x5 - 1},
{5*x1 + 3,
x2 - x5,
25*x5**2 + 5*x5 - 1,
5*x4 + 5*x5 + 1,
5*x3 + 5*x5 + 1},
{5*x1 + 3,
5*x4 - 25*x5**2 + 10*x5 - 2,
x3 - 25*x5**3 + 15*x5**2 - 3*x5,
5*x2 + 125*x5**3 - 50*x5**2 + 10*x5 - 1,
625*x5**4 - 375*x5**3 + 100*x5**2 - 10*x5 + 1},
{5*x1 + 2,
5*x2 + 5*x5 - 1,
5*x4 - 250*x5**3 + 75*x5**2 - 30*x5 + 2,
5*x3 + 250*x5**3 - 75*x5**2 + 30*x5 - 3,
625*x5**4 - 250*x5**3 + 100*x5**2 - 15*x5 + 1},
{x4 + 5*x5**2,
5*x1 + 1,
x3 - 25*x5**3,
5*x2 + 125*x5**3 - 25*x5**2 + 5*x5 - 1,
625*x5**4 - 125*x5**3 + 25*x5**2 - 5*x5 + 1},
{x4 - 5*x5**2,
5*x1 + 4,
x3 - 25*x5**3,
5*x2 + 125*x5**3 + 25*x5**2 + 5*x5 + 1,
625*x5**4 + 125*x5**3 + 25*x5**2 + 5*x5 + 1},
{5*x1 + 3,
5*x2 + 5*x5 + 1,
5*x4 - 250*x5**3 - 75*x5**2 - 30*x5 - 2,
5*x3 + 250*x5**3 + 75*x5**2 + 30*x5 + 3,
625*x5**4 + 250*x5**3 + 100*x5**2 + 15*x5 + 1},
{5*x1 + 2,
5*x4 + 25*x5**2 + 10*x5 + 2,
x3 - 25*x5**3 - 15*x5**2 - 3*x5,
5*x2 + 125*x5**3 + 50*x5**2 + 10*x5 + 1,
625*x5**4 + 375*x5**3 + 100*x5**2 + 10*x5 + 1},
{x5,
x2,
x4,
x3,
x1 + 1},
{x5,
x2,
x4,
x3,
x1}}$
% ==> Testing the linear algebra package
% Find the ideal of points in affine and projective space.
vars:=for k:=1:6 collect mkid(x,k);
vars := {x1,
x2,
x3,
x4,
x5,
x6}$
setring(vars,degreeorder vars,revlex);
{{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$
matrix mm(10,6);
on rounded;
for k:=1:6 do for l:=1:10 do mm(l,k):=floor(exp((k+l)/4));
off rounded;
mm;
mat((1,2,2,3,4,5),(2,2,3,4,5,7),(2,3,4,5,7,9),(3,4,5,7,9,12),(4,5,7,9,12,15),(5,
7,9,12,15,20),(7,9,12,15,20,25),(9,12,15,20,25,33),(12,15,20,25,33,42),(15,20,25
,33,42,54))$
setideal(u,affine_points mm);
{48337*x5**2 - 318*x4*x6 - 75336*x5*x6 + 29579*x6**2 - 11598*x1 - 11016*x2 -
11352*x3 - 2502*x4 + 18371*x5 - 1837*x6 - 1836,
386696*x1*x6 + 175678*x4*x6 + 20108*x5*x6 - 233347*x6**2 - 5821074*x1 - 831000*
x2 + 1382952*x3 + 741984*x4 - 2153934*x5 + 2692351*x6 - 1491936,
386696*x3*x6 + 239238*x4*x6 - 185716*x5*x6 - 182039*x6**2 - 270738*x1 - 1255800*
x2 - 5052384*x3 - 2106864*x4 + 2351946*x5 + 2434483*x6 - 1562736,
48337*x4**2 - 58746*x4*x6 + 148*x5*x6 + 17725*x6**2 + 2502*x1 + 576*x2 - 1302*x3
+ 25721*x4 - 3488*x5 - 12857*x6 + 96,
193348*x4*x5 - 151394*x4*x6 - 118604*x5*x6 + 92869*x6**2 + 1590*x1 + 8712*x2 +
16560*x3 - 3708*x4 + 43918*x5 - 43409*x6 + 1452,
386696*x2*x6 - 382090*x4*x6 - 43364*x5*x6 + 123645*x6**2 - 1313130*x1 - 4809120*
x2 + 91464*x3 + 5853096*x4 + 1118658*x5 - 2323361*x6 - 28128,
193348*x6**3 - 78919578*x4*x6 - 63413004*x5*x6 + 81565689*x6**2 + 1412352942*x1
+ 1563160200*x2 + 761482008*x3 + 10324224*x4 + 304232130*x5 - 1255484065*x6 -
643375200,
193348*x1*x4 - 3606*x4*x6 + 6664*x5*x6 - 36689*x6**2 - 1729374*x1 - 256248*x2 +
422132*x3 + 345556*x4 - 671778*x5 + 747089*x6 - 429404,
193348*x3*x4 - 19782*x4*x6 - 57060*x5*x6 + 1407*x6**2 - 86718*x1 - 378840*x2 -
1475924*x3 - 576996*x4 + 715078*x5 + 671885*x6 - 449836,
386696*x1*x5 + 139942*x4*x6 - 99156*x5*x6 - 94107*x6**2 - 4388370*x1 - 668088*x2
+ 1063040*x3 + 468112*x4 - 1464774*x5 + 1964239*x6 - 1078088,
386696*x3*x5 + 184150*x4*x6 - 329484*x5*x6 + 3733*x6**2 - 302634*x1 - 1062840*x2
- 3845096*x3 - 1562608*x4 + 1956858*x5 + 1752663*x6 - 1143880,
193348*x1**2 + 50726*x4*x6 + 4164*x5*x6 - 49995*x6**2 - 1502518*x1 - 234624*x2 +
337000*x3 + 189344*x4 - 544926*x5 + 709519*x6 - 425800,
193348*x1*x2 - 22834*x4*x6 - 3056*x5*x6 - 4123*x6**2 - 1183010*x1 - 780060*x2 +
282940*x3 + 989552*x4 - 238902*x5 + 102179*x6 - 226684,
386696*x1*x3 + 153254*x4*x6 - 46332*x5*x6 - 109011*x6**2 - 2706290*x1 - 776040*
x2 - 650592*x3 - 288096*x4 - 295470*x5 + 1856303*x6 - 1096080,
96674*x3**2 + 56278*x4*x6 - 44916*x5*x6 - 20423*x6**2 - 85218*x1 - 315900*x2 -
1104614*x3 - 478348*x4 + 559514*x5 + 528787*x6 - 342672,
193348*x2*x4 - 186922*x4*x6 - 13484*x5*x6 + 80823*x6**2 - 400398*x1 - 1437268*x2
+ 32400*x3 + 1825348*x4 + 346526*x5 - 749039*x6 - 13972,
386696*x2*x5 - 299774*x4*x6 - 183476*x5*x6 + 214259*x6**2 - 1032390*x1 - 3818088
*x2 + 38568*x3 + 4563624*x4 + 1175646*x5 - 2005927*x6 - 56304,
193348*x5*x6**2 - 59111766*x4*x6 - 58092824*x5*x6 + 68856463*x6**2 + 1134803514*
x1 + 1224865560*x2 + 591403776*x3 - 36549312*x4 + 316517430*x5 - 1023460331*x6 -
498675720,
96674*x2**2 - 69590*x4*x6 - 7908*x5*x6 + 35327*x6**2 - 243426*x1 - 832910*x2 +
14700*x3 + 1041208*x4 + 204662*x5 - 420851*x6 - 26032,
386696*x2*x3 - 95202*x4*x6 - 92692*x5*x6 + 63421*x6**2 - 736122*x1 - 2670472*x2
- 1669344*x3 + 2061800*x4 + 1466002*x5 - 394913*x6 - 509528,
96674*x4*x6**2 - 29277416*x4*x6 - 19752504*x5*x6 + 28388673*x6**2 + 433544670*x1
+ 473916360*x2 + 235723620*x3 + 39729120*x4 + 97903830*x5 - 411516489*x6 -
188800920}$
setgbasis u$
dim u;
0$
degree u;
10$
setideal(u,proj_points mm);
{457380*x5**3 - 13500*x2*x5*x6 - 20835*x3*x5*x6 + 76950*x4*x5*x6 - 1050234*x5**2
*x6 + 100*x1*x6**2 + 10800*x2*x6**2 + 16568*x3*x6**2 - 60271*x4*x6**2 + 771366*
x5*x6**2 - 179875*x6**3,
330*x4**2 + 1665*x2*x5 + 555*x3*x5 - 4337*x4*x5 - 626*x5**2 + 6*x1*x6 - 1332*x2*
x6 - 450*x3*x6 + 3013*x4*x6 + 2740*x5*x6 - 1635*x6**2,
33*x1*x5 - 90*x2*x5 - 63*x3*x5 + 216*x4*x5 + 60*x5**2 - 25*x1*x6 + 72*x2*x6 + 49
*x3*x6 - 170*x4*x6 - 171*x5*x6 + 97*x6**2,
90*x3**2 - 483*x2*x5 - 183*x3*x5 + 1197*x4*x5 + 174*x5**2 - 2*x1*x6 + 384*x2*x6
+ 68*x3*x6 - 937*x4*x6 - 738*x5*x6 + 485*x6**2,
330*x3*x4 + 555*x2*x5 + 75*x3*x5 - 1519*x4*x5 - 172*x5**2 + 2*x1*x6 - 444*x2*x6
- 260*x3*x6 + 1041*x4*x6 + 950*x5*x6 - 545*x6**2,
457380*x4*x5**2 - 10800*x2*x5*x6 - 9045*x3*x5*x6 - 662625*x4*x5*x6 - 265413*x5**
2*x6 + 80*x1*x6**2 + 8640*x2*x6**2 + 7156*x3*x6**2 + 238408*x4*x6**2 + 391452*x5
*x6**2 - 143900*x6**3,
495*x1*x2 + 1977*x2*x5 + 87*x3*x5 - 4824*x4*x5 - 339*x5**2 - 164*x1*x6 - 1707*x2
*x6 - 97*x3*x6 + 3782*x4*x6 + 2697*x5*x6 - 1840*x6**2,
495*x2**2 + 1134*x2*x5 + 939*x3*x5 - 3771*x4*x5 - 822*x5**2 - 4*x1*x6 - 1257*x2*
x6 - 734*x3*x6 + 2956*x4*x6 + 2709*x5*x6 - 1550*x6**2,
990*x1*x3 - 5043*x2*x5 - 1263*x3*x5 + 12321*x4*x5 + 1866*x5**2 - 464*x1*x6 +
4008*x2*x6 + 788*x3*x6 - 9643*x4*x6 - 7968*x5*x6 + 5165*x6**2,
495*x2*x3 + 1242*x2*x5 - 48*x3*x5 - 3555*x4*x5 - 267*x5**2 + 4*x1*x6 - 1218*x2*
x6 - 157*x3*x6 + 2786*x4*x6 + 2142*x5*x6 - 1420*x6**2,
66*x1*x4 + 345*x2*x5 + 93*x3*x5 - 861*x4*x5 - 120*x5**2 - 38*x1*x6 - 276*x2*x6 -
76*x3*x6 + 659*x4*x6 + 540*x5*x6 - 337*x6**2,
495*x2*x4 + 1770*x2*x5 + 645*x3*x5 - 4908*x4*x5 - 729*x5**2 + 4*x1*x6 - 1713*x2*
x6 - 520*x3*x6 + 3677*x4*x6 + 3165*x5*x6 - 1915*x6**2,
152460*x3*x5**2 + 5880*x2*x5*x6 - 237983*x3*x5*x6 - 6896*x4*x5*x6 - 71438*x5**2*
x6 + 64*x1*x6**2 - 4704*x2*x6**2 + 92748*x3*x6**2 + 5427*x4*x6**2 + 113560*x5*x6
**2 - 45061*x6**3,
990*x1**2 - 1893*x2*x5 + 447*x3*x5 + 3771*x4*x5 + 426*x5**2 - 524*x1*x6 + 1488*
x2*x6 - 322*x3*x6 - 2923*x4*x6 - 2478*x5*x6 + 1715*x6**2,
457380*x2*x5**2 - 754524*x2*x5*x6 - 31530*x3*x5*x6 + 156561*x4*x5*x6 - 132597*x5
**2*x6 + 136*x1*x6**2 + 310896*x2*x6**2 + 25088*x3*x6**2 - 122581*x4*x6**2 +
141732*x5*x6**2 - 30097*x6**3}$
setgbasis u$
dim u;
1$
degree u;
10$
% Change the term order to pure lex in dimension zero.
% Test both approaches, with and without precomputed borderbasis.
vars:=for k:=1:6 collect mkid(x,k);
vars := {x1,
x2,
x3,
x4,
x5,
x6}$
r1:=setring(vars,{},lex);
r1 := {{x1,x2,x3,x4,x5,x6},{},lex,{1,1,1,1,1,1}}$
r2:=setring(vars,degreeorder vars,revlex);
r2 := {{x1,x2,x3,x4,x5,x6},{{1,1,1,1,1,1}},revlex,{1,1,1,1,1,1}}$
setideal(m,{x1**2+x1+2*x2*x6+2*x3*x5+x4**2,
2*x1*x2+x2+2*x3*x6+2*x4*x5,
2*x1*x3+x2**2+x3+2*x4*x6+x5**2,
2*x1*x4+2*x2*x3+x4+2*x5*x6,
2*x1*x5+2*x2*x4+x3**2+x5+x6**2,
2*x1*x6+2*x2*x5+2*x3*x4+x6});
{x1**2 + x4**2 + 2*x3*x5 + 2*x2*x6 + x1,
2*x1*x2 + 2*x4*x5 + 2*x3*x6 + x2,
x2**2 + 2*x1*x3 + x5**2 + 2*x4*x6 + x3,
2*x2*x3 + 2*x1*x4 + 2*x5*x6 + x4,
x3**2 + 2*x2*x4 + 2*x1*x5 + x6**2 + x5,
2*x3*x4 + 2*x2*x5 + 2*x1*x6 + x6}$
gbasis m;
{72*x1*x3*x5*x6 + 36*x3*x5*x6 - 2*x1*x6 - x6,
2*x1*x2 + 2*x4*x5 + 2*x3*x6 + x2,
2*x2*x3 + 2*x1*x4 + 2*x5*x6 + x4,
2*x3*x4 + 2*x2*x5 + 2*x1*x6 + x6,
10368*x6**7 + 5040*x4*x6**4 + 140*x4**2*x6 + 252*x2*x6**2 - 15*x6,
1296*x4*x6**5 + 180*x4**2*x6**2 + 180*x2*x6**3 + 4*x2*x4 - 15*x6**2,
2592*x3*x6**5 - 360*x2*x5*x6**2 + 2*x1*x4 + 12*x5*x6 + x4,
72*x3*x5**2*x6 + 72*x2*x5*x6**2 - 2*x1*x4 - 8*x5*x6 - x4,
36*x2*x5**3 - 108*x1*x4*x6**2 - 72*x5*x6**3 - 54*x4*x6**2 - 5*x3*x6,
18*x4**2*x5 + 18*x3*x5**2 - 18*x1*x6**2 - 9*x6**2 - 2*x5,
36*x4*x5**2 + 36*x4**2*x6 + 72*x3*x5*x6 + 36*x2*x6**2 - 5*x6,
x1**2 + x4**2 + 2*x3*x5 + 2*x2*x6 + x1,
x2**2 + 2*x1*x3 + x5**2 + 2*x4*x6 + x3,
x3**2 + 2*x2*x4 + 2*x1*x5 + x6**2 + x5,
2592*x5*x6**5 + 360*x4*x5*x6**2 + 360*x3*x6**3 - 2*x2*x5 + 10*x1*x6 + 5*x6,
2592*x5**2*x6**3 + 1296*x4*x6**4 + 36*x4**2*x6 + 144*x3*x5*x6 + 180*x2*x6**2 -
13*x6,
72*x5**3*x6 + 216*x4*x5*x6**2 + 72*x3*x6**3 + 2*x2*x5 + 6*x1*x6 + 3*x6,
4*x4**3 - 12*x2*x5**2 - 24*x1*x5*x6 - 4*x6**3 - 12*x5*x6 - x4,
12*x2*x4**2 - 24*x1*x3*x6 - 12*x5**2*x6 - 12*x4*x6**2 - 12*x3*x6 - x2,
1296*x1*x6**5 + 648*x6**5 + 180*x1*x4*x6**2 + 180*x5*x6**3 + 90*x4*x6**2 + x4*x5
+ 6*x3*x6,
2592*x2*x6**5 + 360*x2*x4*x6**2 - 180*x6**4 - 6*x1*x3 - 3*x5**2 - 16*x4*x6 - 3*
x3,
72*x3*x5*x6**3 + 36*x2*x6**4 - x2*x5**2 - 4*x2*x4*x6 - 4*x1*x5*x6 - 6*x6**3 - 2*
x5*x6,
648*x4*x5*x6**3 + 324*x3*x6**4 - 9*x3*x5**2 - 18*x2*x5*x6 + 18*x1*x6**2 + 9*x6**
2 + x5,
2592*x1*x3*x6**3 + 1296*x4*x6**4 + 1296*x3*x6**3 - 36*x4**2*x6 - 72*x3*x5*x6 +
36*x2*x6**2 + 5*x6,
1080*x2*x4*x6**3 + 216*x6**5 - 60*x1*x3*x6 - 30*x5**2*x6 - 90*x4*x6**2 - 30*x3*
x6 - x2,
72*x4**2*x6**3 + 36*x2*x6**4 + 3*x2*x5**2 + 8*x2*x4*x6 + 6*x1*x5*x6 - 2*x6**3 +
3*x5*x6,
36*x1*x5**2*x6 + 36*x1*x4*x6**2 + 36*x5*x6**3 + 18*x5**2*x6 + 18*x4*x6**2 + x4*
x5 + 2*x3*x6,
18*x1*x5**3 - 54*x1*x3*x6**2 - 36*x4*x6**3 + 9*x5**3 - 27*x3*x6**2 + x3*x5 - 2*
x2*x6,
18*x1*x4*x5 + 18*x1*x3*x6 + 18*x5**2*x6 + 18*x4*x6**2 + 9*x4*x5 + 9*x3*x6 + x2,
18*x2*x4*x5 + 18*x1*x5**2 + 18*x1*x4*x6 + 18*x5*x6**2 + 9*x5**2 + 9*x4*x6 + x3,
2*x1*x4**2 + 4*x1*x3*x5 + 2*x5**3 + 8*x4*x5*x6 + 2*x3*x6**2 + x4**2 + 2*x3*x5,
72*x1*x4*x6**3 + 36*x5*x6**4 + 36*x4*x6**3 - 2*x1*x3*x5 - x5**3 - 2*x4*x5*x6 + 2
*x3*x6**2 - x3*x5,
3240*x1*x5*x6**3 + 648*x6**5 + 1620*x5*x6**3 + 90*x1*x3*x6 + 90*x5**2*x6 + 180*
x4*x6**2 + 45*x3*x6 + 2*x2,
72*x2*x5**2*x6 + 72*x2*x4*x6**2 + 144*x1*x5*x6**2 + 36*x6**4 + 72*x5*x6**2 - 2*
x1*x3 - x5**2 - x3,
36*x5**4 - 216*x4**2*x6**2 - 432*x3*x5*x6**2 - 288*x2*x6**3 + 2*x2*x4 + 8*x1*x5
+ 39*x6**2 + 4*x5,
72*x3*x5**3 - 216*x1*x5*x6**2 - 36*x6**4 - 108*x5*x6**2 - 2*x1*x3 - 9*x5**2 - 8*
x4*x6 - x3,
1296*x2*x5*x6**3 + 648*x1*x6**4 + 324*x6**4 - 18*x1*x5**2 - 36*x1*x4*x6 - 72*x5*
x6**2 - 9*x5**2 - 18*x4*x6 - x3,
18*x1*x3*x5**2 + 18*x4**2*x6**2 + 54*x3*x5*x6**2 + 36*x2*x6**3 + 9*x3*x5**2 - x2
*x4 - 2*x1*x5 - 5*x6**2 - x5}$
m1:=change_termorder(m,r1);
m1 := {46656*x4*x5*x6**6 - x4*x5,
80621568*x5*x6**13 + 44928*x5*x6**7 - x5*x6,
637*x2 + 1077507256320*x6**17 - 927987840*x6**11 - 806157*x6**5,
7*x4**2*x5 + 252*x4*x5*x6**3 - 30233088*x5*x6**12 - 17496*x5*x6**6,
58773123072*x6**19 - 47869056*x6**13 - 45657*x6**7 + x6,
2*x1*x4 + x4 + 1296*x5**3*x6**3 + 3*x5*x6,
2548*x4**2*x6 + 91728*x4*x6**4 + 362797056*x6**13 - 33696*x6**7 - 141*x6,
8491392*x4*x6**7 - 182*x4*x6 + 19591041024*x6**16 + 10917504*x6**10 - 243*x6**4,
17199*x5**4*x6 + 1238328*x5**2*x6**5 + 4353564672*x6**15 - 5003856*x6**9 - 1328*
x6**3,
4245696*x5**2*x6**7 - 91*x5**2*x6 + 8707129344*x6**17 - 12130560*x6**11 + 256*x6
**5,
7*x3 - 567*x5**5 - 204120*x5**3*x6**4 + 524880*x5*x6**8 - 90*x5*x6**2,
2916*x5**7 - 567*x5**3*x6**2 + 347680512*x5*x6**12 + 196668*x5*x6**6 - 4*x5,
14*x1*x6 + 504*x4*x5*x6**2 + 252*x5**3*x6 + 20155392*x5*x6**11 + 8640*x5*x6**5 +
7*x6,
2548*x4**3 + 19813248*x4*x6**6 - 637*x4 + 39182082048*x6**15 + 30326400*x6**9 +
4428*x6**3,
1911*x4*x5**2 - 68796*x4*x6**4 - 68796*x5**2*x6**3 - 362797056*x6**13 + 151632*
x6**7 + 50*x6,
1274*x1*x5 + 1651104*x4*x6**5 + 5733*x5**4 + 1238328*x5**2*x6**4 + 637*x5 +
6530347008*x6**14 - 1667952*x6**8 + 192*x6**2,
1274*x1**2 + 1274*x1 + 1274*x4**2 + 206388*x5**6 - 85995*x5**2*x6**2 -
11754624614400*x6**18 + 9498228480*x6**12 + 9295668*x6**6}$
setring r2$
m2:=change_termorder1(m,r1);
m2 := {46656*x4*x5*x6**6 - x4*x5,
80621568*x5*x6**13 + 44928*x5*x6**7 - x5*x6,
2*x1*x4 + x4 + 1296*x5**3*x6**3 + 3*x5*x6,
637*x2 + 1077507256320*x6**17 - 927987840*x6**11 - 806157*x6**5,
7*x4**2*x5 + 252*x4*x5*x6**3 - 30233088*x5*x6**12 - 17496*x5*x6**6,
58773123072*x6**19 - 47869056*x6**13 - 45657*x6**7 + x6,
2548*x4**2*x6 + 91728*x4*x6**4 + 362797056*x6**13 - 33696*x6**7 - 141*x6,
17199*x5**4*x6 + 1238328*x5**2*x6**5 + 4353564672*x6**15 - 5003856*x6**9 - 1328*
x6**3,
4245696*x5**2*x6**7 - 91*x5**2*x6 + 8707129344*x6**17 - 12130560*x6**11 + 256*x6
**5,
7*x3 - 567*x5**5 - 204120*x5**3*x6**4 + 524880*x5*x6**8 - 90*x5*x6**2,
8491392*x4*x6**7 - 182*x4*x6 + 19591041024*x6**16 + 10917504*x6**10 - 243*x6**4,
2916*x5**7 - 567*x5**3*x6**2 + 347680512*x5*x6**12 + 196668*x5*x6**6 - 4*x5,
14*x1*x6 + 504*x4*x5*x6**2 + 252*x5**3*x6 + 20155392*x5*x6**11 + 8640*x5*x6**5 +
7*x6,
1911*x4*x5**2 - 68796*x4*x6**4 - 68796*x5**2*x6**3 - 362797056*x6**13 + 151632*
x6**7 + 50*x6,
2548*x4**3 + 19813248*x4*x6**6 - 637*x4 + 39182082048*x6**15 + 30326400*x6**9 +
4428*x6**3,
1274*x1*x5 + 1651104*x4*x6**5 + 5733*x5**4 + 1238328*x5**2*x6**4 + 637*x5 +
6530347008*x6**14 - 1667952*x6**8 + 192*x6**2,
1274*x1**2 + 1274*x1 + 1274*x4**2 + 206388*x5**6 - 85995*x5**2*x6**2 -
11754624614400*x6**18 + 9498228480*x6**12 + 9295668*x6**6}$
setideal(m1,m1)$
setideal(m2,m2)$
setgbasis m1$
setgbasis m2$
modequalp(m1,m2);
yes$
% ==> Different hilbert series driver
setideal(m,proj_monomial_curve(w1:={0,2,5,9},{w,x,y,z}));
{x**5 - w**3*y**2,
w*y**3 - x**3*z,
y**4 - w*x*z**2,
x**2*y - w**2*z}$
weights:={{1,1,1,1},w1};
weights := {{1,1,1,1},{0,2,5,9}}$
hftestversion 2;
hf!=whilb2$
f1:=weightedhilbertseries(gbasis m,weights);
f1 := ( - w**5*x**17 + w**4*x**17 - w**4*x**15 + w**4*x**8 + w**3*x**15 + w**3*x
**12 + w**3*x**6 + w**2*x**10 + w**2*x**7 + w**2*x**4 + w*x**5 + w*x**2 + 1)/(w
**2*x**9 - w*x**9 - w + 1)$
sub(x=1,ws);
( - w**5 + w**4 + 3*w**3 + 3*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$
% The ordinary Hilbert series.
hftestversion 1;
hf!=whilb1$
% The default.
f2:=weightedhilbertseries(gbasis m,weights);
f2 := ( - w**5*x**17 + w**4*x**17 - w**4*x**15 + w**4*x**8 + w**3*x**15 + w**3*x
**12 + w**3*x**6 + w**2*x**10 + w**2*x**7 + w**2*x**4 + w*x**5 + w*x**2 + 1)/(w
**2*x**9 - w*x**9 - w + 1)$
sub(x=1,ws);
( - w**5 + w**4 + 3*w**3 + 3*w**2 + 2*w + 1)/(w**2 - 2*w + 1)$
f1-f2;
0$
% ==> Different primary decomposition approaches. The example is due
% to Shimoyama Takeshi. CALI 2.2. produced auxiliary embedded
% primes on it.
vars:={dx,dy,x,y};
vars := {dx,dy,x,y}$
setring(vars,degreeorder vars,revlex);
{{dx,dy,x,y},{{1,1,1,1}},revlex,{1,1,1,1}}$
f3:={DY*( - X*DX + Y**2*DY - Y*DY),DX*(X**2*DX - X*DX - Y*DY)}$
primarydecomposition f3;
{{{dx**3,
dy**3,
dx**2*dy,
dx*dy**2,
dy*( - dx*x + dy*y**2 - dy*y),
dx*(dx*x**2 - dx*x - dy*y)},
{dx,dy}},
{{x*y - x - y,
dx*x - dx - dy*y + dy,
- dx + dy*y**2 - 2*dy*y + dy},
{x*y - x - y,
dx*x - dx - dy*y + dy,
- dx + dy*y**2 - 2*dy*y + dy}},
{{dy,x - 1},{dy,x - 1}},
{{dy**2,
dy*x,
x**2,
dx*x + dy*y},
{dy,x}},
{{dx,y - 1},{dx,y - 1}},
{{dx**2,
dx*y,
y**2,
dx*x + dy*y},
{dx,y}},
{{y**2,
x**2,
x*y,
dx*x + dy*y},
{y,x}}}$
showtime;
Time: 6830 ms plus GC time: 140 ms
end;
Time for test: 6830 ms, plus GC time: 150 ms