Schmidt rank and singularities
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 characteris...
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| Date: | 2023 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7227 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512631817764864 |
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| author | Kazhdan, David Lampert, Amichai Polishchuk, Alexander Kazhdan, David Lampert, Amichai Polishchuk, Alexander |
| author_facet | Kazhdan, David Lampert, Amichai Polishchuk, Alexander Kazhdan, David Lampert, Amichai Polishchuk, Alexander |
| author_sort | Kazhdan, David |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:34:38Z |
| description | 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]. |
| doi_str_mv | 10.3842/umzh.v75i9.7227 |
| first_indexed | 2026-03-24T03:31:52Z |
| format | Article |
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| id | umjimathkievua-article-7227 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:52Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-72272024-06-19T00:34:38Z Schmidt rank and singularities Schmidt rank and singularities Kazhdan, David Lampert, Amichai Polishchuk, Alexander Kazhdan, David Lampert, Amichai Polishchuk, Alexander Schmidt rank strength of a polynomial singular locus 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]. УДК 512.5 Ранг Шмідта та сингулярності  Ми знову звертаємося до теореми Шмідта, яка пов'язує ранг Шмідта тензора з корозмірністю певного многовиду, і наводимо доведення, адаптоване до випадку довільної характеристики. Крім того, отримано більш точний результат такого роду для однорідних поліномів за припущення, що характеристика не є дільником степеня. Потім ми використовуємо цей факт, щоб зв’язати ранг Шмідта однорідного полінома (відповідно, колекції однорідних поліномів однакового степеня) з корозмірністю сингулярного локуса відповідної гіперповерхні (відповідно, перетину гіперповерхонь). Це дає ефективну версію теореми Ананяна–Хохстера [J. Amer. Math. Soc., 33, No. 1, 291–309 (2020), Theorem A] Institute of Mathematics, NAS of Ukraine 2023-09-26 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7227 10.3842/umzh.v75i9.7227 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 9 (2023); 1248 - 1266 Український математичний журнал; Том 75 № 9 (2023); 1248 - 1266 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7227/9749 Copyright (c) 2023 Alexander Polishchuk, David Kazhdan |
| spellingShingle | Kazhdan, David Lampert, Amichai Polishchuk, Alexander Kazhdan, David Lampert, Amichai Polishchuk, Alexander Schmidt rank and singularities |
| title | Schmidt rank and singularities |
| title_alt | Schmidt rank and singularities |
| title_full | Schmidt rank and singularities |
| title_fullStr | Schmidt rank and singularities |
| title_full_unstemmed | Schmidt rank and singularities |
| title_short | Schmidt rank and singularities |
| title_sort | schmidt rank and singularities |
| topic_facet | Schmidt rank strength of a polynomial singular locus |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7227 |
| work_keys_str_mv | AT kazhdandavid schmidtrankandsingularities AT lampertamichai schmidtrankandsingularities AT polishchukalexander schmidtrankandsingularities AT kazhdandavid schmidtrankandsingularities AT lampertamichai schmidtrankandsingularities AT polishchukalexander schmidtrankandsingularities |