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minimal set of generators
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2954		 Page 1 of 1
Author:  gstic [ Tue Jun 22, 2021 12:22 am ]
Post subject:  minimal set of generators
minbase and mstd returns a minimal set of generators of an ideal, if the input is homogeneous or if the ordering is local
Given polynomials g=x^3+y^2-1, gx=diff(g, x), gy=diff(g, y) the system of generators for the syzygies
given by Singular is not minimal.
Singular gives for the generators of ideal I=(gx, gy, g):
syz[1]=(2xy, 3y^2-3, -6y)
syz[2]=(-2y, 3x^2, 0)
syz[3]=(x^3+y^2-1, 0, -3x^2)
but a minimal set of generators in this case is
s1=(2xy, 3y^2-3, -6y)
s2=(2x^3-2, 3x^2y, -6x^2)
We can easily check that syz[1], syz[2] and syz[3] can be expressed in term of s1 and s2.
How can I find a minimal set of generators if the ideal is not with homogeneous polynomials
and the ordering is not local ?
Author:  hannes [ Thu Jul 01, 2021 10:10 am ]
Post subject:  Re: minimal set of generators
reply by Douglas Leonard:
For what it is worth,
to produce your results I first used mres to get a syzygy:
ring r=0,(x,y),dp;
> module I=[2xy,3y2-3,-6y],[-2y,3x2,0],[x3+y2-1,0,-3x2];
> resolution mre=mres(I,0);
> mre[2];
_[1]=x2*gen(1)-y2*gen(2)-2y*gen(3)+gen(2)
---------------------------------------------------------
Then I rewrote the syzygy as:
gen(2)=y*(y*gen(2)+2*gen(3))-x^2*gen(1)
---------------------------------------------------
Then I defined:
s_1:=gen(1)
s_2:=y*gen(2)+2*gen(3)
---------------------------------
And finally showed they generated:
gen(1)=s_1
gen(2)=y*s_2-x^2*s_1
2*gen(3)=s_2-y*(y*s_2-x^2*s_1)=(1-y^2)s_2+x^2*y*s_1
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