Reduction and specialization of polynomials (Q2787080)

This paper deals with two major theorems: the Bertini-Noether reduction theorem and Hilbert irreducibility theorem. The main tools are the Grothendieck good reduction criterion in a polynomial context and the introduction of ``\textit{bad primes}''. Let \(A\) be an integral domain. The object of this paper is the irreducibility of polynomials obtained by reduction or specialization from a polynomial \(P\in A[T,Y]\) assumed to be irreducible over the algebraic closure of the field of fractions \(k\) of \(A\). The author introduces the ``\textit{bad prime divisor}'' of \(P\) that somewhat unifies both, the reduction and the specialization questions.NEWLINENEWLINEThe main result isNEWLINENEWLINENEWLINETheorem 3.1: Let \(k\) be a number field. There exist integers \(N, B, C\) and a finite extension \(L/{\mathbb Q}\) such that the following holds: If \(p_1,\ldots,p_N\) are distinct rational primes such that \(p_i\nmid B\), \(p_i\geq C\) and \(p_i\) decomposes fully in \(L/{\mathbb Q}\), then for any multiple \(a\in {\mathbb Z}\) of \(p_1\cdots p_N\), there exists \(b\in{\mathbb N}\) such that the polynomial \(P(am+b,Y)\) is irreducible in \(k[Y]\) for all \(m\in{\mathbb Z}\). Furthermore, \(N,B,C\) and \(L\) can be given explicitly.NEWLINENEWLINENEWLINEThis is presented in Section 3.NEWLINENEWLINEThe new tool, the bad prime divisor of \(P\), is a non-zero parameter \({\mathcal B}_P\in A\) obtained from the coefficients of \(P\) through elementary operations, starting with the discriminant \(\Delta_P\in A[T]\) of \(P\) relative to \(Y\).NEWLINENEWLINEWhen \(A\) is a Dedekind domain of characteristic \(0\), the non-zero prime ideals \({\mathfrak p}\) of \(A\) dividing \({\mathcal B}_P\) are those for which some of the distinct roots of \(\Delta_P\) become equal or infinite modulo \({\mathfrak p}\). Such primes are called bad primes and the others are called good primes.NEWLINENEWLINETheorem 3.1 is an explicit polynomial version of Corollary 4.5 of [\textit{P. Dèbes} and \textit{F. Legrand}, Adv. Stud. Pure Math. 63, 141--162 (2012; Zbl 1321.11114)]. It is a first application of the other main result:NEWLINENEWLINETheorem 2.6. This theorem establishes that if \({\mathfrak p}\) is a good prime such that \(\deg_Y P!\notin {\mathfrak p}\), then (a) \({\mathcal B}_{P\bmod {\mathfrak p}} ={\mathcal B}_P\bmod {\mathfrak p}\) and is non-zero in \(A/{\mathfrak p}\); (b) the polynomial \(P\) modulo \({\mathfrak p}\) is irreducible in \(\bar{\kappa}_{\mathfrak p} [T,Y]\), where \(\kappa_{\mathfrak p}\) is the field of fractions of \(A/{\mathfrak p}\) (``Bertini-Noether''); (c) if in addition \({\mathfrak p}\) is of large norm and some other assumption holds, then each element of the Galois group \({\mathcal G}\) of \(P\) over \(\bar{k}(T)\) is the Frobenius at \({\mathfrak p}\) of the splitting field of some specialization \(P(t_0,Y)\), \(t_0\in A\) (``Chebotarev'').NEWLINENEWLINEThe proof of Theorem 2.6 is postponed to last section.