On Hilbert’s irreducibility theorem

Abstract

Abstract. In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if f (X, T 1 , . . . , Ts) is an irreducible polynomial with integer coefficients, having Galois group G over the function field Q(T 1 , . . . , Ts), and K is any subgroup of G, then there are at most O f,ε (H s−1+|G/K| −1 +ε ) specialisations t ∈ Z s with |t| ≤ H such that the resulting polynomial f (X) has Galois group K over the rationals.

Effective Hilbert's Irreducibility Theorem for global fields

“…This bound is new in the case when K ≠ Q, whereas when K = Q, this improves the bound given in [7]. Before ending this introduction, we must mention that the dependence on the field K can be made explicit in all of the bounds.…”

“…We are now in position to state the Bombieri-Pila type of bound that we have obtained in [17]. [2,6] log(H K (P d )) + d [3,7] log(B) + d [4,8] , d [4, 14 3 ] b(P )} H K (P d ) [4,8] ≲ K d [4,8] (log(H K,aff (P ))…”

“…By the same argument as in the proof of Claim 4.3, and by inequality (5.5), we conclude that for all v ∈ M K,∞ , (5.6) [26,35] deg T (P H ) [21,26] (log(H K,aff (P H )) + 1) [6,10] B 28,37] (log(H K (F )) + 1) [6,10] The case when F is non monic is dealt as in the proof of Theorem While the order of magnitude on T of the bound of Theorem 1.3 is in general sharp as can be seen by considering the polynomial F (T, Y ) = Y 2 − T , in some cases one may obtain better bounds. Indeed, when K = Q in [7] Castillo and Dietmann use Galois resolvents to find an adequate primitive element for the splitting field L of F , and use [6, Theorem 1] (which is a Bombieri-Pila type of bound over Q for lopsided boxes) to deduce a variant of Hilbert's irreducibility theorem for Galois groups that takes into consideration the subgroup structure of the Galois group of the specialized polynomial. By means of Theorem 1.2 and a lemma about Galois resolvents in [7], we improve and generalize [7,Theorem 1] to number fields, and we improve upon Theorem 1.3 in the case when K is a number field.…”

Differential elimination for dynamical models via projections with applications to structural identifiability

“…In the proof of the proposition, we will use the following quantitative version of the Hilbert irreducibility theorem due to Cohen [12] (see also [10]). Lemma 4.7 (Follows from [12, Theorem 2.1]).…”

Irreducibility criteria for compositions of multivariate polynomials over arbitrary fields

“…One of the most influential results in the study of multivariate polynomials and their specializations is the famous Hilbert Irreducibility Theorem [35]. There are many fundamental results on this subject obtained in the last decades, and here we will only refer the reader to the works of Castillo and Dietmann [15], Cavachi [16], Corvaja [18],…”