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Grigoriev and Podolskii (2018) have established a tropical analogue of theeffective Nullstellensatz, showing that a system of tropical polynomialequations is solvable if and only if a linearized system obtained from atruncated Macaulay matrix is solvable. They provided an upper bound of theminimal admissible truncation degree, as a function of the degrees of thetropical polynomials. We establish a tropical Nullstellensatz adapted to {\emsparse} tropical polynomial systems. Our approach is inspired by a constructionof Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improvedbound of the truncation degree, which coincides with the classical Macaulaydegree in the case of $n+1$ equations in $n$ unknowns. We also establish atropical Positivstellensatz, allowing one to decide the inclusion of tropicalbasic semialgebraic sets. This allows one to reduce decision problems fortropical semi-algebraic sets to the solution of systems of tropical linearequalities and inequalities.

The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems