Reduction and specialization of polynomials

Pierre Dèbes¹

¹ Laboratoire Paul Painlevé Mathématiques Université de Lille 59655 Villeneuve d'Ascq Cedex, France

Acta Arithmetica, Tome 172 (2016) no. 2, pp. 175-197

Résumé

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 $P(x_1,\ldots,x_s,T,Y)$.

Détail

DOI : 10.4064/aa8176-12-2015

Keywords: explicit forms bertini noether reduction theorem hilbert irreducibility theorem approach recasts polynomial context geometric grothendieck reduction criterion congruence approach hit covers line notion bad primes polynomial mathbb irreducible overline mathbb introduced which plays central unifying role polynomial deduce bound least integer geq irreducible mathbb generic which galois group overline mathbb deg bound only depends degree number bad primes similar issues addressed algebraic families polynomials ldots y