Probabilistic Galois theory of reciprocal polynomials (Q1130134)

A polynomial \(f(X)\in \mathbb{Q}[X]\) is called reciprocal if \(f(X)= X^nf(1/X)\). Let \(n= 2m\) be even. The Galois group of a reciprocal polynomial is a subgroup of the semidirect product of \((\mathbb{Z}/2\mathbb{Z})^m\) with the symmetric group \({\mathfrak S}_m\) on \(m\) letters. Denote the latter group by \({\mathcal G}\). The purpose of the paper under review is to show that the Galois group of a randomly chosen reciprocal polynomial is likely to be \({\mathcal G}\), hence as large as possible. More precisely, we get the following result. Let \({\mathcal E}(N)\) be the number of reciprocal polynomials \(f\) of degree \(2m\) with Galois group a proper subgroup of \({\mathcal G}\) and such that any coefficient of \(f\) is an integer between \(-N\) and \(N\). Then \[ {\mathcal E}(N)\ll N^{m- 1/2}\log N \] with the implied constant depending only on \(m\). The main tool in the proof is a multidimensional form of the large sieve inequality.