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). This gives an effective version of Ananyan-Hochster's Theorem A from arXiv:1610.09268.
Keywords
Cite
@article{arxiv.2104.10198,
title = {Schmidt rank and singularities},
author = {David Kazhdan and Amichai Lampert and Alexander Polishchuk},
journal= {arXiv preprint arXiv:2104.10198},
year = {2023}
}
Comments
v1: 17 pages; v2: 19 pages, added Theorem 1.5 on generic derivatives; v3: improved exposition, added references; v4: 20 pages, added Amichai Lampert as a coauthor, improved bounds in the main theorems