Description:

Title: (Quasi-)linear bounds for rank in algebraic closure

Abstract: For polynomials of fixed degree in many variables, rank is a fundamental quantity measuring non-degeneracy. Over algebraically closed fields, rank is closely related to singularities and can thus be used to estimate exponential sums. However, rank can decrease when passing to an algebraic closure, and in applications - such as to higher order Fourier analysis and the Hardy-Littlewood circle method - it is important to bound this potential decrease. For many fields, including number fields and function fields, we prove that strength decreases by at most a constant factor (depending on the degree and the field). This partially resolves a conjecture of Adiprasito, Kazhdan and Ziegler. For finite fields, our results combine with those of Moshkovitz-Zhu to obtain a state-of-the-art quasi-linear bound.
Joint work with Benjamin Baily.