Let $K$ be a field. Often, the problem of solving an equation over the polynomial ring $K[X]$ can be translated into solving a system of equations over $K$. For example, consider the following property a polynomial $f \in K[X]$ of degree $2m$ can have.
There exist $f_1, f_2, f_3 \in K[X]$ of degree at most $m$ such that $f = f_1^2 + f_2^2 + f_3^2$.
Writing $f_j = \sum_{i=0}^m A_i^{(j)}X^i$ for each $j \in \lbrace 1, 2, 3 \rbrace$, the above property can be translated into the solvability over $K$ of a system of $2m+1$ equations in the $3m+3$ variables $(A_i^{(j)})_{i, j}$.
I am interested in whether this system is (absolutely) irreducible. That is, if we assume $K$ is algebraically closed, whether the equations defining this system generate a prime ideal of the polynomial ring in the variables $(A_i^{(j)})_{i, j}$. How could one approach this problem?
There exists algorithms to decide whether an ideal of a polynomial ring is prime or not.
You can find one in Section 4.4. of ADAMS, LOUSTAUNAU, An introduction to Gröbner bases, Grad. Maths. Texts 3 .
The algorithm is described on p.244.