- Series
- Number Theory
- Time
- Wednesday, September 18, 2024 - 3:30pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Evan O'Dorney – Carnegie Mellon University – emo916math@gmail.com – https://sites.google.com/view/evan-m-odorney
- Organizer
- Alexander Dunn

Given a random polynomial f of degree n with integer coefficients each drawn uniformly and independently from an interval [-H, H], what is the probability that the Galois group of the roots of f is NOT the full symmetric group Sₙ? In 1936, van der Waerden conjectured that the answer should be of order 1/H, with the dominant contribution coming from f with a rational root. This conjecture was finally resolved by Bhargava in 2023. In this project (joint w/ Theresa Anderson), we ask the same question for reciprocal (a.k.a. palindromic) polynomials, which arise for instance as the characteristic polynomials of symplectic matrices. Using a suitably modified variant of the Fourier-analytic methods of Bhargava and others, we find that polynomials with non-generic Galois group appear with frequency O(log H/H) and, unlike in van der Waerden's setting, almost all of these polynomials are irreducible.