Schmidt rank/strength and the singular locus

Seminar
Algebra
Speaker
Amichai Lampert (University of Michigan)
Date
03/01/2024 - 11:30 - 10:30Add to Calendar 2024-01-03 10:30:00 2024-01-03 11:30:00 Schmidt rank/strength and the singular locus The Schmidt rank/strength of a polynomial is an algebraic measure of its non-degeneracy. It has proven very useful for studying questions regarding polynomials of fixed degree in arbitrarily many variables: Schmidt used it to count integer solutions for systems of polynomial equations with rational coefficients, Green and Tao used it to investigate the distribution of values of polynomials over finite fields, and Ananyan and Hochster used it to prove Stillman's conjecture on projective dimension of ideals in polynomial rings.  A central tool in all these applications is a relationship between Schmidt rank/strength of a polynomial and a geometric measure of its non-degeneracy - the codimension of the singular locus of the polynomial. I will present a recent result on quantitative bounds for this relationship and discuss some related results and questions. Joint work with David Kazhdan and Alexander Polishchuk.  ================================================ https://us02web.zoom.us/j/87856132062 Meeting ID: 878 5613 2062 Third floor seminar room, Mathematics building, and on Zoom. See link below. אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room, Mathematics building, and on Zoom. See link below.
Abstract

The Schmidt rank/strength of a polynomial is an algebraic measure of its non-degeneracy. It has proven very useful for studying questions regarding polynomials of fixed degree in arbitrarily many variables: Schmidt used it to count integer solutions for systems of polynomial equations with rational coefficients, Green and Tao used it to investigate the distribution of values of polynomials over finite fields, and Ananyan and Hochster used it to prove Stillman's conjecture on projective dimension of ideals in polynomial rings.  A central tool in all these applications is a relationship between Schmidt rank/strength of a polynomial and a geometric measure of its non-degeneracy - the codimension of the singular locus of the polynomial. I will present a recent result on quantitative bounds for this relationship and discuss some related results and questions.


Joint work with David Kazhdan and Alexander Polishchuk. 

================================================

https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062

Last Updated Date : 27/12/2023