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Polynomials with Squarefree Discriminant

Introductory Workshop: Diophantine Geometry February 06, 2023 - February 10, 2023

February 10, 2023 (03:30 PM PST - 04:30 PM PST)
Speaker(s): Arul Shankar (University of Toronto)
Location: MSRI: Simons Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

Polynomials With Squarefree Discriminant


A classical question in analytic number theory is: given a polynomial with integer coefficients, how often does it take squarefree values? In arithmetic statistics, we are particularly interested in the case of discriminant polynomials. In this talk, I will present several different cases of this question. First, we will consider a classical result of Davenport--Heilbronn which considers the case of discriminants of binary cubic forms. Motivated by this case, I will discuss two works, joint with Manjul Bhargava and Xiaoheng Wang, in which we consider the spaces of degree-n polynomials and degree-n binary forms. We will prove that a positive proportion of these polynomials and forms have squarefree discriminant.

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Polynomials With Squarefree Discriminant

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