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Tropical Combinatorial Nullstellensatz and Sparse Polynomials

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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:

  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.

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Notes

  1. For two nonnegative real-valued functions f(k, n) and g(n, k), the notation \(f = \varTheta (g)\) means that there are positive constants c and C such that \(cf(k,n) \le g(k,n) \le Cf(k,n)\) for all k and n.

  2. For the sake of completeness, we show how to deduce this fact from Lemma 5.1 in [34]. If the number of rows in the system is less than the number of columns in it, we add several copies of one of equations to make the matrix of the system square. Clearly, this square matrix is singular, and by Lemma 5.1 in [34] (applied to the transposed matrix), the points whose coordinates form the rows of the matrix lie on some tropical hyperplane. The coefficients of this hyperplane form a solution to our linear system.

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Acknowledgements

We would like to thank the reviewers for numerous helpful comments.

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Authors and Affiliations

  1. CNRS, Mathématiques, Université de Lille, 59655, Villeneuve d’Ascq, France

    Dima Grigoriev

  2. Steklov Mathematical Institute, 8 Gubkina St., Moscow, Russia, 119991

    Vladimir V. Podolskii

  3. National Research University Higher School of Economics, Moscow, Russia

    Vladimir V. Podolskii

Authors
  1. Dima Grigoriev
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  2. Vladimir V. Podolskii
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Correspondence to Vladimir V. Podolskii.

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Communicated by Teresa Krick.

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An extended abstract of a preliminary version [17] appeared in the proceedings of the 21st International Symposium on Fundamentals of Computation Theory (FCT 2017). The results of Sects. 4 and 6 were obtained by the first author at MCCME and supported by the Russian Science Foundation (Project 16-11-10075). The results of Sects. 3 and 5 were obtained by the second author and were supported by grant MK-5379.2018.1, by the Russian Academic Excellence Project ‘5-100’ and by RFBR Grant 17-51-10005-KO_a.

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Grigoriev, D., Podolskii, V.V. Tropical Combinatorial Nullstellensatz and Sparse Polynomials. Found Comput Math 20, 753–781 (2020). https://doi.org/10.1007/s10208-019-09431-1

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