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.



Software to compute generators of a module over polynomial ring
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

```

