Schmidt rank and singularities - Kazhdan, David; Lampert, Amichai; Polishchuk, Alexander (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 512.5We 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, assuming the characteristic does not divide the degree. Further, we 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 [J. Amer. Math. Soc., 33, No. 1, 291–309 (2020), Theorem A].