Mathematics Colloquium
Date: Friday, 11 April 2025
Time: 3:30 to 4:30 PM
Venue: Seminar Hall, CMI
Polynomials with squarefree discriminant
Arul Shankar
University of Toronto.
11-04-25

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Abstract

A classical problem in number theory is: what is the density of squarefree values taken by an integer polynomial? In general, it is not unconditionally known that this density exists, or even that the lower density is positive. However, when the polynomial in question has certain additional structure, this may be exploited to answer the question completely.

In this talk, we will study this problem for the case of certain discriminant polynomials. In particular, we will prove that when degree-n polynomials are ordered by the size of their coefficients, a positive proportion of them have squarefree discriminants. We will then see various applications of these results to arithmetic statistics. This is joint work with Manjul Bhargava and Xiaoheng Wang.

This talk will be accessible to general audience.