Reducibility of polynomials f(x,y) modulo p

الأصل · EN

We consider absolutely irreducible polynomials f ∈ Z[x,y] with °ₓ(f)=m, °y(f)=n and height H. We show that for any prime p with p>cmn H2mn+n-1 the reduction f p is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials f ∈ Z[x,y] which are reducible mod p where p is a prime with p>H²ᵐ.