Relative rank and regularization
We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation betwe...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942400015X/type/journal_article |
| _version_ | 1827325437531914240 |
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| author | Amichai Lampert Tamar Ziegler |
| author_facet | Amichai Lampert Tamar Ziegler |
| author_sort | Amichai Lampert |
| collection | DOAJ |
| description | We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials
$\mathcal P=(P_i)_{i=1}^c$
of degrees
$\le d$
in a polynomial ring over an algebraically closed field of characteristic
$>d$
is contained in an ideal
$\mathcal I({\mathcal Q})$
, generated by a collection
${\mathcal Q}$
of polynomials of degrees
$\le d$
which form a regular sequence, and
${\mathcal Q}$
is of size
$\le A c^{A}$
, where
$A=A(d)$
is independent of the number of variables. |
| first_indexed | 2024-03-07T14:27:11Z |
| format | Article |
| id | doaj.art-828e01c83fd14c54a1bdca7a579d6f42 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-07T14:27:11Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-828e01c83fd14c54a1bdca7a579d6f422024-03-06T05:58:15ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.15Relative rank and regularizationAmichai Lampert0https://orcid.org/0000-0002-4386-7275Tamar Ziegler1Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA; E-mail:Einstein Institute of Mathematics, Edmond J. Safra Campus, Hebrew University of Jerusalem, Givat Ram 91904, Jerusalem, Israel; E-mail:We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials $\mathcal P=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal I({\mathcal Q})$ , generated by a collection ${\mathcal Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and ${\mathcal Q}$ is of size $\le A c^{A}$ , where $A=A(d)$ is independent of the number of variables.https://www.cambridge.org/core/product/identifier/S205050942400015X/type/journal_article11T0611B3013F20 |
| spellingShingle | Amichai Lampert Tamar Ziegler Relative rank and regularization Forum of Mathematics, Sigma 11T06 11B30 13F20 |
| title | Relative rank and regularization |
| title_full | Relative rank and regularization |
| title_fullStr | Relative rank and regularization |
| title_full_unstemmed | Relative rank and regularization |
| title_short | Relative rank and regularization |
| title_sort | relative rank and regularization |
| topic | 11T06 11B30 13F20 |
| url | https://www.cambridge.org/core/product/identifier/S205050942400015X/type/journal_article |
| work_keys_str_mv | AT amichailampert relativerankandregularization AT tamarziegler relativerankandregularization |