Schmidt rank and singularities

2021/04/20 by David Kazhdan, Kazhdan, David, Lampert, Amichai +1 · 3 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2104.10198

openalex publication_date 2021/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Cited by

• Polynomial Bounds for Birch's Theorem
• A weak regularity lemma for polynomials
• Slice rank and partition rank of the determinant

Related

• Waring rank of symmetric tensors, and singularities of some projective hypersurfaces
• The symmetric geometric rank of symmetric tensors
• Geometric lower bounds for generalized ranks
• Quasi-linear relation between partition and analytic rank
• Strength is bounded linearly by Birch rank