Appears in collection : Combinatorics and Arithmetic for Physics: special days 2023

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 polynomi-als. We establish a tropical Nullstellensatz adapted to sparse tropical polynomial systems. Our approach is inspired by a polyhedral 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 derive a tropical Positivstellensatz, al-lowing one to decide the inclusion of tropical basic semialgebraic sets. We finally show that solutions can be computed by a reduction to parametric mean-payoff games, providing a tropical analogue of eigenvalue methods to solve polynomial systems. This is a joint work with Marianne Akian and Antoine Bereau.