Discrete & Computational Geometry

, Volume 59, Issue 3, pp 507–552 | Cite as

Tropical Effective Primary and Dual Nullstellensätze

  • Dima Grigoriev
  • Vladimir V. PodolskiiEmail author


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.


Nullstellensatz Min-plus algebra Tropical algebra 

Mathematics Subject Classification

14T05 68W30 52C99 



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. 1.CNRS, MathématiquesUniversité de LilleVilleneuve-d’AscqFrance
  2. 2.Steklov Mathematical InstituteMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia

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