Finding (scalar) relations among polynomials

[Steven Sam]

I have a collection of 2521 polynomials of degree 16 in six variables.  When I use mingens, Macaulay2 tells me after about 1 minute that there are 55 scalar relations, i.e., some subset of 2466 of the polynomials span the whole set. I would like to know exactly what these linear dependencies are. The command // does not finish, even if I ask it to express just 1 polynomial in terms of the 2466. Is there something else that I can do, i.e., can mingens spit out more information if suitably modified? It seems that this problem is not too complicated for Macaulay2 given that mingens tells me nontrivial information.

Motivation: I obtained these polynomials by substituting certain quartics into a collection of quartic monomials. I wanted to know what degree 4 relations these quartics satisfied. Apparently there are 55 of them, but I want to know what they are.

Thanks,
Steven

[Brian Pike]

Hi Steven,
Could you use syz to compute these relations?  If you're only interested in linear relations, then maybe you can limit the computation by setting the option DegreeLimit=>1?  (See the help page for gb for other options; Algorithm and SyzygyLimit look interesting.)

For example, try running:
R=QQ[x,y,z];
m=matrix{{2*x,3*y,x*y*z}};
-- this gives no linear relations:
syz(m,DegreeLimit=>1)
-- this gives one degree <2 relation:
syz(m,DegreeLimit=>2)
-- this gives two relations of degree <3:
syz(m,DegreeLimit=>3)

n=matrix{{2*x,3*x,x*y*z}};
-- For this set, we have one linear relation:
syz(n,DegreeLimit=>1)

You could also try your original set of polynomials, and use syz with DegreeLimit=>5.  I don't know if that would be any easier.

If that doesn't help, you could always translate your problem into linear algebra by taking a basis of the space of degree 16 polynomials in six variables ((16+6-1)!/(16!*(5!))=20349 dimensional).  It would be a very large system, but if it were sparse enough then it might be reasonable.  (I don't know that M2 would be the correct program to use here.)

Thanks,
Brian Pike

[Steven Sam]

Hi Brian, the syz command works perfectly, thanks!