I am interested to know whether factorization of a multivariate polynomial $f(x_1, x_2, \dots, x_n) \in \mathbb{F}_p[x_1, x_2, \dots, x_n]$ into irreducible factors yields some information about the roots of $f(x_1, x_2, \dots, x_n)$.
For instance, for the univariate polynomials we know that if $f(x) = (x-a)(x-b) = 0$, then $x=a, x=b$ are the roots of the polynomial $f(x)$. Is there a similar relation between the irreducible factors of a multivariate polynomial $f(x_1, x_2, \dots, x_n)$ in $\mathbb{F}_p[x_1, x_2, \dots, x_n]$ and its roots? I would really appreciate any literature suggestions on this topic as well, thank you!