List of algorithm problems in term of ideals
We all know that 3SAT is NP-complete. Here is how to reduce 3SAT to your problem.

Suppose that the 3SAT instance has variables $x_1,\ldots,x_n$ and clauses $C_1,\ldots,C_m$. The new instance will have variables $x_1,\ldots,x_n,y_1,\ldots,y_m,z_1,\ldots,z_m$, and the following polynomials:


$x_i^2 - x_i$ for each $i \in [n]$ and $y_j^2 - y_j, z_j^2 - z_j$ for each $j \in [m]$.
For a clause $C_\ell$ of the form $x_i \lor x_j \lor x_k$: $x_i + x_j + x_k - y_\ell - z_\ell - 1$.
If a clause has negated variables, say $x_i$ appears as $\overline{x_i}$, then replace $x_i$ with $1-x_i$ in the polynomial stated above.


Why does this work, the first type of polynomials constrain all variables to be $0,1$. The polynomial $x_i + x_j + x_k - y_\ell - z_\ell - 1$ states that $x_i + x_j + x_k = 1 + y_\ell + z_\ell$, and since $y_\ell,z_\ell$ occur only in this polynomial, this is the same as $1 \leq x_i + x_j + x_k \leq 3$, that is, $x_i + x_j + x_k \geq 1$, or equivalently, $x_i \lor x_j \lor x_k$.