Schmidt rank and singularities

David Kazhdan; Amichai Lampert; Alexander Polishchuk

doi:10.3842/umzh.v75i9.7227

David Kazhdan Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
Amichai Lampert Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel and Institute for Advanced Study, Princeton, NJ, USA
Alexander Polishchuk Department of Mathematics, University of Oregon, Eugene, USA, National Research University Higher School of Economics and Korea Institute for Advanced Study

DOI: https://doi.org/10.3842/umzh.v75i9.7227

Keywords: Schmidt rank, strength of a polynomial, singular locus

Abstract

UDC 512.5

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, 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].

References

K. Adiprasito, D. Kazhdan, T. Ziegler, On the Schmidt and analytic ranks for trilinear forms/em>; arXiv:2102.03659.

T. Ananyan, M. Hochster, Small subalgebras of polynomial rings and Stillman's conjecture, J. Amer. Math. Soc., 33, № 1, 291–309 (2020). DOI: https://doi.org/10.1090/jams/932

T. Ananyan, M. Hochster, Strength conditions, small subalgebras, and Stillman bounds in degree $le 4$, Trans. Amer. Math. Soc., 373, № 7, 4757–4806 (2020). DOI: https://doi.org/10.1090/tran/8060

E. Ballico, A. Bik, A. Oneto, E. Ventura, Strength and slice rank of forms are genericaly equal; arXiv:2102.11549.

E. Ballico, A. Bik, A. Oneto, E. Ventura, The set of forms with bounded strength is not closed; arXiv:2012.01237.

J. M. Boardman, Singularities of differentiable maps, Publ. Math. Inst. Hautes 'Edutes Sci., 33, 383–419 (1967). DOI: https://doi.org/10.1007/BF02684585

A. D. R. Choudary, A. Dimca, On the dual and Hessian mappings of projective hypersurfaces, Math. Proc. Camb. Phil. Soc., 101, 461–468 (1987). DOI: https://doi.org/10.1017/S0305004100066834

H. Derksen, The $G$-stable rank for tensors; arXiv:2002.08435.

D. Kazhdan, T. Ziegler, On the codimension of the singular locus; arXiv:1907.11750.

S. Kopparty, G. Moshkovitz, J. Zuiddam, Geometric rank of tensors and subrank of matrix multiplication; arXiv:2002.09472.

W. M. Schmidt, The density of integer points on homogeneous varieties, Acta Math., 154, № 3-4, 243–296 (1985). DOI: https://doi.org/10.1007/BF02392473

F. Zak, Structure of Gauss maps, Funct. Anal. and Appl., 21, 32–41 (1987). DOI: https://doi.org/10.1007/BF01077983

A. Cohen, G. Moshkovitz, Partition and analytic rank are equivalent over large fields; arXiv:2102.10509.