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 regarding projective dimension of ideals in polynomial rings. A central tool in all these applications is a close 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.