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Tropical Combinatorial Nullstellensatz and Fewnomials Testing

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Fundamentals of Computation Theory (FCT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10472))

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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 an important 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 three 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 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 non-zero polynomial with at most k monomials has a non-root among these inputs?

In the classical algebra well-known results in the direction of these questions are Combinatorial Nullstellensatz, Schwartz-Zippel Lemma and Universal Testing Set for sparse polynomials respectively. In this paper we extensively study these three questions for tropical polynomials and provide results analogous to the classical results mentioned above.

The first author is grateful to the grant RSF 16-11-10075 and to MCCME for inspiring atmosphere. The work of the second author is partially supported by the grant of the President of Russian Federation (MK-7312.2016.1) and by the Russian Academic Excellence Project ‘5–100’.

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Author information

Authors and Affiliations

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

    Dima Grigoriev

  2. Steklov Mathematical Institute, Moscow, Russia

    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 .

Editor information

Editors and Affiliations

  1. CNRS, University of Bordeaux, Talence cedex, France

    Ralf Klasing

  2. LaBRI, University of Bordeaux, Talence cedex, France

    Marc Zeitoun

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Grigoriev, D., Podolskii, V.V. (2017). Tropical Combinatorial Nullstellensatz and Fewnomials Testing. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_23

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  • DOI: https://doi.org/10.1007/978-3-662-55751-8_23

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