The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems

Explore a 46-minute lecture on the tropical analogue of the effective Nullstellensatz and Positivstellensatz for sparse polynomial systems. Delve into the work of Grigoriev and Podolskii, who established that a system of tropical polynomial equations is solvable if and only if a linearized system from a truncated Macaulay matrix is solvable. Discover an improved bound of the truncation degree for sparse tropical polynomial systems, inspired by the polyhedral construction of Canny-Emiris and refined by Sturmfels. Learn about the tropical Positivstellensatz and its application in deciding the inclusion of tropical basic semialgebraic sets. Examine how solutions can be computed through a reduction to parametric mean-payoff games, providing a tropical analogue of eigenvalue methods for solving polynomial systems. Gain insights from this joint work by Marianne Akian, Antoine Bereau, and Stéphane Gaubert, presented by Stéphane Gaubert from INRIA and CMAP, Ecole polytechnique, IP Paris, CNRS at the Institut des Hautes Etudes Scientifiques (IHES).