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Tropical Effective Primary and Dual Nullstellensätze

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Abstract

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows studying properties of mathematical objects such as algebraic varieties from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz, and moreover, we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.

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Notes

  1. To avoid a confusion we note that we use the word “dual” in two different meanings. First, we use it in the term “dual Nullstellensatz” as opposed to the standard version of Nullstellensatz. This means that the dual Nullstellensatz is obtained from the standard Nullstellensatz by the (linear) duality (see [11] and later on in this paper). Second, we use the word “dual” in the term “duality result” to denote the general type of results. Since the standard Nullstellensatz is a duality result itself, applying the linear duality to it results in a non-duality result. Thus, the dual Nullstellensatz is not a duality result in a proper sense, but rather the word “dual” is used in contrast to the customary Nullstellensatz which we name “primary”.

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Acknowledgements

The first author is grateful to the Grant RSF 16-11-10075 and to both MCCME and the Max-Planck Institut für Mathematik, Bonn for wonderful working conditions and an inspiring atmosphere.

The work of the second author is partially supported by the grant of the President of Russian Federation (MK-5379.2018.1) and by the Russian Academic Excellence Project ‘5-100’. Part of the work of the second author was done during the visit to Max-Planck Institut für Mathematik, Bonn.

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Authors and Affiliations

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

    Dima Grigoriev

  2. Steklov Mathematical Institute, 119991, Moscow, Russia

    Vladimir V. Podolskii

  3. National Research University Higher School of Economics, 125319, 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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Editor in Charge János Pach

An extended abstract of a preliminary version [14] appeared in the proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015).

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Grigoriev, D., Podolskii, V.V. Tropical Effective Primary and Dual Nullstellensätze. Discrete Comput Geom 59, 507–552 (2018). https://doi.org/10.1007/s00454-018-9966-3

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  • DOI: https://doi.org/10.1007/s00454-018-9966-3

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