TY - JOUR
T1 - Schmidt Rank and Singularities
AU - Kazhdan, David
AU - Lampert, Amichai
AU - Polishchuk, Alexander
N1 - Publisher Copyright:
© Springer Science+Business Media, LLC, part of Springer Nature 2024.
PY - 2024/2
Y1 - 2024/2
N2 - 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 establish a sharper result for this kind for homogeneous polynomials, assuming that 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].
AB - 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 establish a sharper result for this kind for homogeneous polynomials, assuming that 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].
UR - http://www.scopus.com/inward/record.url?scp=85185307762&partnerID=8YFLogxK
U2 - 10.1007/s11253-024-02270-6
DO - 10.1007/s11253-024-02270-6
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AN - SCOPUS:85185307762
SN - 0041-5995
VL - 75
SP - 1420
EP - 1442
JO - Ukrainian Mathematical Journal
JF - Ukrainian Mathematical Journal
IS - 9
ER -