Tropical Combinatorial Nullstellensatz and Sparse Polynomials

Abstract

Tropical 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. Tropical polynomials play a fundamental role in this, especially for the case of algebraic geometry. On the other hand, many algebraic questions behind tropical polynomials remain open. In this paper, we address four basic questions on tropical polynomials closely related to their computational properties: In the classical algebra well-known results in the direction of these questions are Combinatorial Nullstellensatz due to N. Alon, J. Schwartz–R. Zippel Lemma and Universal Testing Set for sparse polynomials, respectively. The classical analog of the last question is known as $$\tau $$ -conjecture due to M. Shub–S. Smale. In this paper, we provide results on these four questions for tropical polynomials.