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 $(x_1,\dots,x_n)\not\equiv0\bmod m$ of $r\geq n$ homogeneous $n$ variable polynomials
$$f_1(x_1,\dots,x_n)\equiv0\bmod m$$
$$f_2(x_1,\dots,x_n)\equiv0\bmod m$$
$$\vdots$$
$$f_r(x_1,\dots,x_n)\equiv0\bmod m$$
with $m$ of unknown factorization 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 $O((\log m)^n)$).