The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems

Marianne Akian (1) , Antoine Béreau (1) , Stéphane Gaubert (1)

1 TROPICAL - TROPICAL

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 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.

Mots clés: Tropical geometry; Polynomial systems; Newton polytope; Tropical hypersurfaces; Combinatorics of convex polytopes

Domaines: Mathématiques [math]

Marianne Akian, Antoine Béreau, Stéphane Gaubert. The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. 2023. ⟨hal-04333931v2⟩