Relative Rank and Regularization

Abstract

We introduce a new notion of rank -relative rank associated to a filtered collection of polynomials. When the filtration consists of one set the notion coincides with the Schmidt rank (also called strength). We also introduce the notion of relative bias. We prove a relation between these two notions over finite fields. This allows us to perform a regularization procedure in a number of steps that is polynomial in the initial number of polynomials. As an application we prove that any collection of homogeneous polynomials P = (P i ) c i=1 of degrees ≤ d in a polynomial ring over an algebraically closed field of characteristic > d is contained in an ideal I(Q), generated by a collection Q of homogeneous polynomials of degrees ≤ d which form a regular sequence, and Q is of size ≤ Ac A , where A = A(d) is independent of the number of variables.

The Geometry of Polynomial Representations

“…Indeed, over finite fields K of characteristic > d, the strength of a degree-d form f is closely related to its analytic rank, which measures the statistical bias of f regarded as a function K n → K. This line of research goes back to Green and Tao [GT], a polynomial upper bound for strength in terms of analytic rank was found by Milićević [M], and this was further improved by Cohen, Moshkovitz, and Zhu to (almost) linear bounds [CM, MZ]. The connection between strength and bias has been exploited to establish further algebraic properties of high-strength forms by Kazhdan,Lampert,Polishchuk,and Ziegler [KaZ,KLP,LZ1,LZ2]. Interestingly, this is an entirely different route to such algebraic properties than the one we take here.…”