(SOLVED) Testing a system of polynomials for roots over $\mathbb{F}_p$

(Alex Ozdemir)

See solution: Algorithm for testing a system of polynomials for roots over $\mathbb F_p$,

Given $n$ $m$-variate polynomials $P_1, \ldots, P_n$ of degree at most $d$, with coefficients in $\mathbb{Z}/p \mathbb{Z}$, is there an algorithm for determining whether $P_1, \ldots, P_n$ have any common roots in $(\mathbb{Z}/p\mathbb{Z})^m$ in time polynomial in $\operatorname{log} p$? The algorithm’s runtime dependence on $n$, $m$, and $d$ is not important. Note that exhaustive search takes at least $p$ time.

Follow-up: Is there an algorithm that runs in polynomial time in $n$, $m$, $d$, and $\log p$?