Tropical Effective Primary and Dual Nullstellensätze

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 importa...

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Bibliographic Details
Published in:Discrete & computational geometry 2018-04, Vol.59 (3), p.507-552
Main Authors: Grigoriev, Dima, Podolskii, Vladimir V.
Format: Article
Language:English
Subjects:
Algebra
Combinatorics
Computation
Computational Mathematics and Numerical Analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
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Summary: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.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-018-9966-3