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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.08410 |
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| _version_ | 1866909484545212416 |
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| author | Bihan, Frédéric Dickenstein, Alicia Forsgård, Jens |
| author_facet | Bihan, Frédéric Dickenstein, Alicia Forsgård, Jens |
| contents | Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparse systems with high local multiplicity Bihan, Frédéric Dickenstein, Alicia Forsgård, Jens Algebraic Geometry Commutative Algebra Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions. |
| title | Sparse systems with high local multiplicity |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2402.08410 |