Max-plus polynomials and their roots

Max-plus algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects computationally accessible. Max-plus polynomials play a fundamental role in this, but on the other hand, many questions on properties of max-plus polynomials remain open. In this talk we will discuss some recent results on max-plus polynomials (of many variables) and their roots. In particular, we will discuss max-plus analogs of Combinatorial Nullstellensatz, Schwartz-Zippel Lemma and Universal Testing Set. If there is enough time, we will also discuss max-plus Hilbert's Nullstellensatz.