Generic consistency and nondegeneracy of vertically parametrized systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912624071933952 |
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| author | Feliu, Elisenda Henriksson, Oskar Pascual-Escudero, Beatriz |
| author_facet | Feliu, Elisenda Henriksson, Oskar Pascual-Escudero, Beatriz |
| contents | We determine the generic consistency, dimension and nondegeneracy of the zero locus over $\mathbb{C}^*$, $\mathbb{R}^*$ and $\mathbb{R}_{>0}$ of vertically parametrized systems: parametric polynomial systems consisting of linear combinations of monomials scaled by free parameters. These systems generalize sparse systems with fixed monomial support and freely varying parametric coefficients. As our main result, we establish the equivalence among three key properties: the existence of nondegenerate zeros, the zero set having generically the expected dimension, and the system being generically consistent. Importantly, we prove that checking whether a vertically parametrized system has these properties amounts to an easily computed matrix rank condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02302 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic consistency and nondegeneracy of vertically parametrized systems Feliu, Elisenda Henriksson, Oskar Pascual-Escudero, Beatriz Algebraic Geometry Molecular Networks Quantitative Methods 13P15, 14A10, 14A25, 14Q30 We determine the generic consistency, dimension and nondegeneracy of the zero locus over $\mathbb{C}^*$, $\mathbb{R}^*$ and $\mathbb{R}_{>0}$ of vertically parametrized systems: parametric polynomial systems consisting of linear combinations of monomials scaled by free parameters. These systems generalize sparse systems with fixed monomial support and freely varying parametric coefficients. As our main result, we establish the equivalence among three key properties: the existence of nondegenerate zeros, the zero set having generically the expected dimension, and the system being generically consistent. Importantly, we prove that checking whether a vertically parametrized system has these properties amounts to an easily computed matrix rank condition. |
| title | Generic consistency and nondegeneracy of vertically parametrized systems |
| topic | Algebraic Geometry Molecular Networks Quantitative Methods 13P15, 14A10, 14A25, 14Q30 |
| url | https://arxiv.org/abs/2304.02302 |