# Tropical Combinatorial Nullstellensatz and Sparse Polynomials.

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:

1. Given a polynomial with a certain support (set of monomials) and a (finite) set of inputs, when is it possible for the polynomial to vanish on all these inputs?

2. A more precise question, given a polynomial with a certain support and a (finite) set of inputs, how many roots can polynomial have on this set of inputs?

3. Given an integer $k$, for which $s$ there is a set of $s$ inputs such that any non-zero polynomial with at most $k$ monomials has a non-root among these inputs?

4. How many integer roots can have a one variable polynomial given by an tropical algebraic circuit?

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.

Publisher URL: http://arxiv.org/abs/1706.00080

DOI: arXiv:1706.00080v2

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.