Let $p_1,\dotsc,p_m$ be real polynomials (although rational can do as well) in $t$ variables.

What is the most efficient way to compute the smallest degree $d$ such that $s(p_1,\dotsc,p_m) = 0$ for some rational symmetric polynomial $s$ in $m$ variables?

At the moment I am doing this for $t = 1$ by iterating the following simple algorithm over $d \geq 1$:

Create a list

`Sc`

of the Schur functions corresponding to the partitions of $k \leq d$ with at most $m$ parts, which is a linear basis over $\Bbb{Q}$ for the space of symmetric polynomials in $m$ variables with degree up to $d$.Compute the list

`Ps`

of the evaluations of each element of`Sc`

at $(p_1,\dotsc,p_m)$.Convert the elements of

`Ps`

into vectors and stop if they are linearly dependent, otherwise increase $d$ by one and repeat from 1.

This works reasonably well when the degrees of $p_1,\dotsc,p_m$ are small, but otherwise iterating over each $d$ involves quite a bit of work and it doesn't scale very well.

I know that the equivalent problem for $s \in \Bbb{Q}[X_1,\dotsc,X_m]$ is "readily" solved through variable elimination via Gröbner bases, but I have yet to find a way to make this work for symmetric polynomials. I also thought that I might compute the relevant ideal and then try to find the subset fixed by the symmetric group in $m$ variables, but I couldn't find any facilities in Sage to compute fixed sets under a group action (which I guess is a hard problem in general).