Title: Constructing Large Galois Groups via Small Perturbations
Abstract: A classical theorem of van der Waerden shows that, in a density sense, most integer polynomials of a fixed degree have the full symmetric group as their Galois group. However, this result is non-constructive and does not explain how to explicitly produce such polynomials close to a given one.
In this talk, I will present a constructive approach to this problem. Given an integer polynomial $ f(x) $ of degree $d $, we show that there exists a nearby polynomial $g(x) $, differing from $ f(x) $ by a bounded perturbation, such that the Galois group of $g(x) $ over $\mathbb{Q} $ is $ S_d $.
The method relies on explicit coefficient constructions, Newton polygon techniques, local ramification criteria, and global group-theoretic arguments to control ramification and enforce large Galois groups. It also incorporates a root-distribution argument using Rouché’s theorem and the recent resolution of the Schinzel–Zassenhaus conjecture by Dimitrov.
This talk is based on ongoing joint work with Pradipto Banerjee.It will be held via Zoom in room D2 314.