The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems.
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| Title: | The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. |
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| Authors: | Akian, Marianne1,2 (AUTHOR) marianne.akian@inria.fr, Béreau, Antoine1,2 (AUTHOR) antoine.bereau@ac-creteil.fr, Gaubert, Stéphane1,2 (AUTHOR) stephane.gaubert@inria.fr |
| Source: | Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1587-1633. 47p. |
| Subjects: | Semialgebraic sets, Algebraic geometry, Mathematical inequalities, Matrices (Mathematics) |
| Abstract: | Grigoriev and Podolskii (Discrete Comput Geom 59:507–552, 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 and Emiris (in: Applied algebra, algebraic algorithms and error-correcting codes. Proceedings of 10th international symposium, AAECC-10, San Juan de Puerto Rico, Springer, Berlin, 1993), refined by Sturmfels (J Algebr Combin 3(2):207–236, 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. [ABSTRACT FROM AUTHOR] |
| Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 194201085 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Akian%2C+Marianne%22">Akian, Marianne</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> marianne.akian@inria.fr</i><br /><searchLink fieldCode="AR" term="%22Béreau%2C+Antoine%22">Béreau, Antoine</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> antoine.bereau@ac-creteil.fr</i><br /><searchLink fieldCode="AR" term="%22Gaubert%2C+Stéphane%22">Gaubert, Stéphane</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> stephane.gaubert@inria.fr</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Jun2026, Vol. 26 Issue 3, p1587-1633. 47p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Semialgebraic+sets%22">Semialgebraic sets</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+geometry%22">Algebraic geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+inequalities%22">Mathematical inequalities</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Grigoriev and Podolskii (Discrete Comput Geom 59:507–552, 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 and Emiris (in: Applied algebra, algebraic algorithms and error-correcting codes. Proceedings of 10th international symposium, AAECC-10, San Juan de Puerto Rico, Springer, Berlin, 1993), refined by Sturmfels (J Algebr Combin 3(2):207–236, 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. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10208-025-09708-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 47 StartPage: 1587 Subjects: – SubjectFull: Semialgebraic sets Type: general – SubjectFull: Algebraic geometry Type: general – SubjectFull: Mathematical inequalities Type: general – SubjectFull: Matrices (Mathematics) Type: general Titles: – TitleFull: The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Akian, Marianne – PersonEntity: Name: NameFull: Béreau, Antoine – PersonEntity: Name: NameFull: Gaubert, Stéphane IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 3 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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