Probabilistic Galois Theory

TL;DR

This paper establishes a new upper bound on the number of monic integer polynomials of degree n with bounded height that do not have the full symmetric group as their Galois group, improving previous estimates.

Contribution

It provides an improved upper bound on the count of polynomials with non-maximal Galois groups, advancing the understanding of Galois group distribution for integer polynomials.

Findings

New upper bound: $O_{n,\epsilon}(H^{n-2+\sqrt{2}+\epsilon})$

Improvement over previous bound: $O_{n}(H^{n-1/2}\log H)$

Enhanced understanding of Galois group distribution

Abstract

We show that there are at most $O_{n,\epsilon}(H^{n-2+\sqrt{2}+\epsilon})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_{n}(H^{n-1/2}\log H)$.