Reducibility of polynomials $f(x,y)$ modulo $p$

Abstract

We consider absolutely irreducible polynomials $f\in Z[x,y]$ with ${°}_x(f)=m$, ${°}_y(f)=n$ and height $H$. We show that for any prime $p$ with $p>c_{mn}{H}^{2mn+n-1}$ the reduction $f\mathrm{mod}p$ is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials $f\in Z[x,y]$ which are reducible mod $p$ where $p$ is a prime with $p>H^{2m}$.

Latex, 7 pages