Schmidt rank and singularities

Abstract

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials of degree d (assuming that the characteristic does not divide d(d − 1)). We then use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces).

Schmidt rank and algebraic closure

“…Kazhdan and Polishchuk extended a result of Schmidt and showed a relationship between the Birch singular locus and the Schmidt rank for a multilinear form P over a general field k [9].…”

“…Cohen and Moshkovitz [3] proved a similar result for a single multi-linear polynomial. Here is theorem 1.4 from [9].…”

“…there exists a point y ∈ Y such that the Jacobian matrix has rank n at y. But then the tangent space at y has codimension n, contradicting the fact that the local codimension at y is < n. Kazhdan and Polishchuk [9], extending results of Schmidt, showed that a similar inequality holds also for higher degrees. Cohen and Moshkovitz [3] proved a similar result for a single multi-linear polynomial.…”

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.…”