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.

If $r\geq n$ homogeneous polynomials given have $n$ variables are of form $$f_1(x_1,\dots,x_n)\equiv0\bmod m$$ $$f_2(x_1,\dots,x_n)\equiv0\bmod m$$ $$\vdots$$ $$f_r(x_1,\dots,x_n)\equiv0\bmod m$$ where $m$ is of unknown factorization can resultants reveal all common root $(x_1,\dots,x_n)\not\equiv0\bmod m$ or is there different technique that applies?