Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

We show explicit forms of the Bertini–Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of ‶bad primes″ of a polynomial $P\in \mathbb Q[T,Y]$ irreducible over $\overline{\mathbb{Q}}$ is introduced, which plays a central and unifying role. For such a polynomial $P$, we deduce a new bound for the least integer $t_0\geq 0$ such that $P(t_0,Y)$ is irreducible in $\mathbb{Q}[Y]$: in the generic case for which the Galois group of $P$ over $\overline{\mathbb{Q}}(T)$ is $S_n$ ($n=\deg_Y(P)$), this bound only depends on the degree of $P$ and the number of bad primes. Similar issues are addressed for algebraic families of polynomials $P(x_1,\ldots,x_s,T,Y)$.