roots of bivariate polynomial over prime field

Question

We know any polynomial with degree n over real field has at most n roots. Let $p(x)$ is a bivariate polynomial with degree $n$ over prime field $F_p$. How many roots existe over $F_p$ ? If $p(x)$ be a univariate polynomial, what is answer?

Answer 1

I say: if $p(x)$ be a univariate polynomial then has at most $p^n$ roots. Since $p(x)=a_0+a_1*x+...+a_{n-1}*x^n$ s.t $a_i$s are in $F_p$ and any $a_i$ has $p$ state. Also if $p(x)$ be a bivariate polynomial then has at most $p^{n(n+1)/2}$ roots