Schmidt rank and singularities

Author:

Kazhdan David,Lampert Amichai,Polishchuk Alexander

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., <strong>33</strong>, No. 1, 291–309 (2020), Theorem A].

Publisher

SIGMA (Symmetry, Integrability and Geometry: Methods and Application)

Reference13 articles.

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

2. T. Ananyan, M. Hochster, Small subalgebras of polynomial rings and Stillman's conjecture, J. Amer. Math. Soc., 33, № 1, 291–309 (2020).

3. T. Ananyan, M. Hochster, Strength conditions, small subalgebras, and Stillman bounds in degree $le 4$, Trans. Amer. Math. Soc., 373, № 7, 4757–4806 (2020).

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

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

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3