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 $r\geq n+1$ homogeneous $n+1$ variable polynomials given have form 
$$f_1(x_0,x_1,\dots,x_n)\equiv0\bmod m$$
$$f_2(x_0,x_1,\dots,x_n)\equiv0\bmod m$$
$$\vdots$$
$$f_r(x_0,x_1,\dots,x_n)\equiv0\bmod m$$
where $m$ is prime can resultants reveal all common root $(1,x_1,\dots,x_n)\not\equiv(1,0,\dots,0)\bmod m$ or is there different technique that applies?