The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| author | Akian, Marianne Béreau, Antoine Gaubert, Stéphane |
| author_facet | Akian, Marianne Béreau, Antoine Gaubert, Stéphane |
| contents | Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay matrix is solvable. They provided an upper bound of the minimal admissible truncation degree, as a function of the degrees of the tropical polynomials. We establish a tropical Nullstellensatz adapted to {\em sparse} tropical polynomial systems. Our approach is inspired by a construction of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved bound of the truncation degree, which coincides with the classical Macaulay degree in the case of $n+1$ equations in $n$ unknowns. We also establish a tropical Positivstellensatz, allowing one to decide the inclusion of tropical basic semialgebraic sets. This allows one to reduce decision problems for tropical semi-algebraic sets to the solution of systems of tropical linear equalities and inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05859 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems Akian, Marianne Béreau, Antoine Gaubert, Stéphane Combinatorics Algebraic Geometry 14T15 (Primary) 05E14, 14N10, 52B20 (Secondary) Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay matrix is solvable. They provided an upper bound of the minimal admissible truncation degree, as a function of the degrees of the tropical polynomials. We establish a tropical Nullstellensatz adapted to {\em sparse} tropical polynomial systems. Our approach is inspired by a construction of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved bound of the truncation degree, which coincides with the classical Macaulay degree in the case of $n+1$ equations in $n$ unknowns. We also establish a tropical Positivstellensatz, allowing one to decide the inclusion of tropical basic semialgebraic sets. This allows one to reduce decision problems for tropical semi-algebraic sets to the solution of systems of tropical linear equalities and inequalities. |
| title | The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems |
| topic | Combinatorics Algebraic Geometry 14T15 (Primary) 05E14, 14N10, 52B20 (Secondary) |
| url | https://arxiv.org/abs/2312.05859 |