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arXiv · 2312.05859

The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems

Marianne Akian·Antoine Béreau·Stéphane Gaubert
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Abstract

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.

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Marianne Akian, Antoine Béreau, Stéphane Gaubert. 2023-12-10. The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. https://arxiv.org/abs/2312.05859

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