Macaulay Resultants help identify if $n$ homogeneous polynomials in $n$ variables have a common root in $\mathbb P^{n-1}(\mathbb K)$ when $\mathbb K$ is algebraically closed. Is it possible resultants reveal all common roots $(x_0,x_1,\dots,x_n)\not\equiv(1,0,\dots,0,0)\bmod m$ of $r\geq n+1$ homogeneous $n+1$ variable polynomials $$f_1(x_0,x_1,\dots,x_n)\equiv0\bmod m$$ $$f_2(x_0,x_1,\dots,x_n)\equiv0\bmod m$$ $$\vdots$$ $$f_r(x_0,x_1,\dots,x_n)\equiv0\bmod m$$ with $m$ prime or is there different technique that applies (we assume the coefficient matrix is of rank $r$)?

The method could be exponential in $n$ (note Macaulay Resultants are exponential in $n$ and I only seek a method that is compatible in complexity with algebraically closed case). It only needs to depend polylogarithmically in $m$ (it is trivial to find a method with $O(m^n)$ complexity but I seek something $O((\log m)^n)$).