Ideals of Spaces of Degenerate Matrices
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866915808159989760 |
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| author | Vill, Julian Michałek, Mateusz Blomenhofer, Alexander Taveira |
| author_facet | Vill, Julian Michałek, Mateusz Blomenhofer, Alexander Taveira |
| contents | The variety $ \mathrm{Sing}_{n, m} $ consists of all tuples $ X = (X_1,\ldots, X_m) $ of $ n\times n $ matrices such that every linear combination of $ X_1,\ldots, X_m $ is singular. Equivalently, $X\in\mathrm{Sing}_{n,m}$ if and only if $\det(λ_1 X_1 + \ldots + λ_m X_m) = 0$ for all $ λ_1,\ldots, λ_m\in \mathbb Q $. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $ \mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials $\det(λ_1 X_1 + \ldots + λ_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $ \mathrm{Sing}_{2, m} $ for all $ m\in \mathbb N $. Our results exhibit that there are additional equations arising from the tensor structure of $ X $. More generally, for any $ n $ and $ m\ge n^2 - n + 1 $, we prove there are equations vanishing on $ \mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of type $\det(λ_1 X_1 + \ldots + λ_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_00735 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Ideals of Spaces of Degenerate Matrices Vill, Julian Michałek, Mateusz Blomenhofer, Alexander Taveira Commutative Algebra Algebraic Geometry Representation Theory 15A69, 14Q20 The variety $ \mathrm{Sing}_{n, m} $ consists of all tuples $ X = (X_1,\ldots, X_m) $ of $ n\times n $ matrices such that every linear combination of $ X_1,\ldots, X_m $ is singular. Equivalently, $X\in\mathrm{Sing}_{n,m}$ if and only if $\det(λ_1 X_1 + \ldots + λ_m X_m) = 0$ for all $ λ_1,\ldots, λ_m\in \mathbb Q $. Makam and Wigderson asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on $ \mathrm{Sing}_{n, m} $ lies in the ideal generated by the polynomials $\det(λ_1 X_1 + \ldots + λ_m X_m)$. We answer this question in the negative by determining the vanishing ideal of $ \mathrm{Sing}_{2, m} $ for all $ m\in \mathbb N $. Our results exhibit that there are additional equations arising from the tensor structure of $ X $. More generally, for any $ n $ and $ m\ge n^2 - n + 1 $, we prove there are equations vanishing on $ \mathrm{Sing}_{n, m} $ that are not in the ideal generated by polynomials of type $\det(λ_1 X_1 + \ldots + λ_m X_m)$. Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases. |
| title | Ideals of Spaces of Degenerate Matrices |
| topic | Commutative Algebra Algebraic Geometry Representation Theory 15A69, 14Q20 |
| url | https://arxiv.org/abs/2106.00735 |