The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems - Centre de mathématiques appliquées (CMAP)
Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems

Marianne Akian
  • Fonction : Auteur
Antoine Béreau
Stéphane Gaubert

Résumé

Grigoriev and Podolskii (2018) have established a tropical analogue of the effective Nullstellensatz, showing that a system of tropical polynomial equations is solvable if and only if a linearized system obtained from a truncated Macaulay matrix is solvable. They provided an upper bound of the minimal admissible truncation degree, as a function of the degrees of the tropical polynomials. We establish a tropical Nullstellensatz adapted to sparse tropical polynomial systems. Our approach is inspired by a construction of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved bound of the truncation degree, which coincides with the classical Macaulay degree in the case of n + 1 equations in n unknowns. We also establish a tropical Positivstellensatz, allowing one to decide the inclusion of tropical basic semialgebraic sets. This allows one to reduce decision problems for tropical semi-algebraic sets to the solution of systems of tropical linear equalities and inequalities.
Fichier principal
Vignette du fichier
article-tropical-nullstellensatz.pdf (583.52 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04333931 , version 1 (10-12-2023)
hal-04333931 , version 2 (21-10-2024)

Licence

Copyright (Tous droits réservés)

Identifiants

Citer

Marianne Akian, Antoine Béreau, Stéphane Gaubert. The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. 2023. ⟨hal-04333931v2⟩
49 Consultations
20 Téléchargements

Altmetric

Partager

More