Intersection of two polynomials in $F_p[x]$

Question

The number of intersections of two polynomials of degrees $m$ and $n$ in $R$ is at most $max(m, n)$.

Is this true in $F_p[x]$, being $p$ a large prime? How can it be proved?

Answer 1

Just following up from the comments. The result is true for a simpler reason that the fundamental theorem of algebra, which doesn't apply to $F_p[x]$. The factor theorem. Let $R$ be an integral domain, then $R[x]$ possesses a division algorithm. Indeed, for all $f,g \in R[x]$, and $\alpha \in R$ the leading coefficient of $G$. Then there exists some integer $i$ and some $q,r \in R[x]$ with $0 \leq \deg{r} < \deg{g}$ such that $\alpha^i f = qg + r$. In particular suppose that $f(a) = 0$, and let $g = (x-a)$. Then there exists some $q,r \in R[x]$ with $f = (x-a)q + r$ such that $0 \leq \deg{r} < \deg{x-a} = 1$, and so $\deg{r} = 0$ and so $r \in R$. But then $0 = f(a) = (a-a)q(a) + r = r$, and so $r = 0$. Hence $f = (x-a)q$, but then clearly if $f$ has distinct roots it has most $\deg{f}$ roots, and so $f$ always has at most $\deg{f}$ roots.