Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Abstract

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$ with $\mathrm{Gal}(f)\subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.

Minor revisions

References in corpus (3)

• Hilbert's irreducibility theorem and the larger sieve
• Upper bounds on number fields of given degree and bounded discriminant
• Towards van der Waerden's conjecture

Cited by in corpus (3)

• Galois groups of random additive polynomials
• Irreducibility of the characteristic polynomials of random tridiagonal matrices
• Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem