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. 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. 2. A more precise question, given a polynomial with a certain support and a (finite) set of inputs, how many roots can this polynomial have on this set of inputs?
3. 3. Given an integer k, for which s there is a set of s inputs such that any nonzero polynomial with at most k monomials has a non-root among these inputs?
4. 4. How many integer roots can have a one variable polynomial given by a 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.

中文翻译：

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热带组合Nullstellensatz和稀疏多项式

热带代数出现在数学的许多领域，例如代数几何，数学物理和组合优化。它的重要性部分与以下事实有关：它使数学对象的各种参数在计算上可访问。热带多项式在其中起着基本作用，尤其是对于代数几何而言。另一方面，热带多项式背后的许多代数问题仍未解决。在本文中，我们针对与热带热带多项式的计算特性密切相关的四个基本问题：

1. 1. 给定具有一定支持（多项式集合）和一组（有限）输入项的多项式，何时多项式对所有这些输入项消失？
2. 2. 一个更精确的问题，给定一个具有一定支持和一组（有限）输入的多项式，该多项式在这组输入上可以有多少个根？
3. 3. 给定一个整数k，对于s有一组s输入，以至于任何具有最多k个单项式的非零多项式在这些输入中都具有非根？
4. 4. 一个热带代数回路可以给出一个变量多项式有多少个整数根？

在经典代数中，针对这些问题的著名结果是N. Alon，J. Schwartz–R提出的组合Nullstellensatz。稀疏多项式的Zippel引理和通用测试集。由于M. Shub–S ，最后一个问题的经典比喻被称为\（\ tau \）-猜想。人妖 在本文中，我们提供了关于热带多项式这四个问题的结果。