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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2406.13658 |
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| author | DiPasquale, Michael Fouli, Louiza Kumar, Arvind Tohǎneanu, Ştefan O. |
| author_facet | DiPasquale, Michael Fouli, Louiza Kumar, Arvind Tohǎneanu, Ştefan O. |
| contents | It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for $r$). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids DiPasquale, Michael Fouli, Louiza Kumar, Arvind Tohǎneanu, Ştefan O. Commutative Algebra 94B05, 05B35, 05E40, 13F55, 51E10 It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for $r$). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design. |
| title | Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids |
| topic | Commutative Algebra 94B05, 05B35, 05E40, 13F55, 51E10 |
| url | https://arxiv.org/abs/2406.13658 |