Toric invariance of vertically parametrized systems
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917494710599680 |
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| author | Feliu, Elisenda Henriksson, Oskar |
| author_facet | Feliu, Elisenda Henriksson, Oskar |
| contents | We consider the problem of deciding whether the solution sets of a parametrized polynomial system are toric in the sense that they admit a monomial parametrization. We focus on vertically parametrized systems, which are sparse systems where we allow linear dependencies between coefficients in front of the same monomial. We give a matroid-theoretic characterization of the maximal-dimensional torus for which all solution sets are invariant under componentwise multiplication. Building on this, we provide necessary conditions and sufficient conditions for when the solution sets are unions of finitely many or a unique coset of this torus. The motivation of this work comes from the theory of reaction networks, where toric structure of the steady state system substantially simplifies the determination of multistationarity; here, we show that this is also the case for absolute concentration robustness and steady state invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric invariance of vertically parametrized systems Feliu, Elisenda Henriksson, Oskar Algebraic Geometry 14M25, 14A10, 14A25, 14Q30, 14P10 We consider the problem of deciding whether the solution sets of a parametrized polynomial system are toric in the sense that they admit a monomial parametrization. We focus on vertically parametrized systems, which are sparse systems where we allow linear dependencies between coefficients in front of the same monomial. We give a matroid-theoretic characterization of the maximal-dimensional torus for which all solution sets are invariant under componentwise multiplication. Building on this, we provide necessary conditions and sufficient conditions for when the solution sets are unions of finitely many or a unique coset of this torus. The motivation of this work comes from the theory of reaction networks, where toric structure of the steady state system substantially simplifies the determination of multistationarity; here, we show that this is also the case for absolute concentration robustness and steady state invariants. |
| title | Toric invariance of vertically parametrized systems |
| topic | Algebraic Geometry 14M25, 14A10, 14A25, 14Q30, 14P10 |
| url | https://arxiv.org/abs/2411.15134 |