Software to compute generators of a module over polynomial ring

Question

Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.

Consider the subset $$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}.$$ Clearly $M$ is an $A$-submodule of $A^k$. Necessarily $M$ is finitely generated.

I am wondering if there exists a software which allows to compute explicit generators of $M$ as an $A$-module.

In my case $n=16,k=10$, and all $p_i$'s are explicit homogeneous polynomials of second degree.

Answer 1

What you compute is the "syzygy module" of $p_1,\ldots,p_k$.

You can try the following M2 script to check the computation times in Macaulay 2 for different $n$'s and $k$'s.

For me $n=16, k=6$ was a matter of minutes, but $k > 6$ seems to take much longer.

If you decrease the sparsity of your $p_i$ by setting Density to 0.1, 0.5 or higher, you will probably be out of luck with your computation.

n = 16
k = 10
R=QQ
--- or R=ZZ/23 or any other prime
A=R[x_1..x_n]
mat = random(A^1, A^{k:-2}, Density=>0.05)
erz = gens ker mat