On a tropical dual Nullstellensatz
type: Item
instance of: scholarly article
title: On a tropical dual Nullstellensatz
zbMATH Open document ID: 1247.14067
published in: Advances in Applied Mathematics
publication date: 2012-03-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.0519
reviewed by: Péter E. Frenkel
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 6019338
zbMATH Keywords: dual Nullstellensatz
zbMATH Keywords: solving tropical polynomial systems
zbMATH Keywords: Cayley matrix
author: Dima Grigoriev
MaRDI profile type: Publication
OpenAlex ID: W1977220030
arXiv ID: 1108.0519
cites work: Max-linear Systems: Theory and Algorithms
cites work: Q5290261
cites work: On the efficiency of effective Nullstellensätze
cites work: Tropical algebraic geometry
cites work: TROPICAL ALGEBRAIC SETS, IDEALS AND AN ALGEBRAIC NULLSTELLENSATZ
cites work: The Tropical Rank of a Tropical Matrix
cites work: Sharp Effective Nullstellensatz
cites work: A tropical nullstellensatz
cites work: Tropical resultants for curves and stable intersection
cites work: On the frontiers of polynomial computations in tropical geometry
DOI: 10.1016/J.AAM.2011.11.004
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review text: Consider \(s\) finite, convex sequences of real numbers, i.e., we have \(a_{j,k}\) for \(j=1,\dots, s\); \(k=0,1,\dots, d_j\), such that \(a_{j,k-1}+a_{j,k+1}\geq 2a_{j,k}\). Put \(N=4\sum d_j\) and \(D=\max d_j\). Suppose that there exists a sequence of real numbers \(y_k\) for \(-N\leq k\leq N+D\) such that for all \(|i|\leq N\) and all \(j\), the minimum \(\min_k (y_{k+i}+a_{j,k})\) is attained for at least two different values of \(k\in\{0,1,\dots, d_j\}\). The paper under review proves that then \((y_k)\) can be chosen to be an arithmetic progression.NEWLINENEWLINEIt is explained why it is appropriate to call this a tropical effective dual Nullstellensatz for tropical univariate polynomials. The sequences \((a_{j,k})\) can be thought of as a system of \(s\) tropical polynomials. The existence of \((y_k)\) can be thought of as the solvability of the tropical linear system given by the Cayley matrix of that tropical polynomial system. The existence of a suitable arithmetic progression \((y_k)\) can be thought of as solvability of the polynomial system itself.NEWLINENEWLINEIt is pointed out that the result for \(s=2\) follows from the approach of [\textit{L. F. Tabera}, Rev. Mat. Iberoam. 24, No. 3, 941--961 (2008; Zbl 1216.14048)], which relies on a theorem due to Kapranov, but that approach does not work for general \(s\).NEWLINENEWLINEA multivariate generalization is conjectured. It is pointed out that the direct tropical analog of the Nullstellensatz fails already in the univariate case and therefore it is natural to study the dual version.
description: scientific article; zbMATH DE number 6019338