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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| Published in | Discrete & computational geometry Vol. 59; no. 3; pp. 507 - 552 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.04.2018
Springer Nature B.V Springer Verlag |
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| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-018-9966-3 |
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| Abstract | 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. |
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| AbstractList | 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. 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 to study properties of mathematical objects such as algebraic varieties and algebraic curves 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 minplus polynomial systems. |
| Author | Podolskii, Vladimir V. Grigoriev, Dima |
| Author_xml | – sequence: 1 givenname: Dima surname: Grigoriev fullname: Grigoriev, Dima organization: CNRS, Mathématiques, Université de Lille – sequence: 2 givenname: Vladimir V. orcidid: 0000-0001-7154-138X surname: Podolskii fullname: Podolskii, Vladimir V. email: podolskii@mi.ras.ru organization: Steklov Mathematical Institute, National Research University Higher School of Economics |
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| CitedBy_id | crossref_primary_10_1007_s10208_019_09431_1 crossref_primary_10_1016_j_aam_2020_102012 crossref_primary_10_1016_j_aam_2023_102635 crossref_primary_10_1016_j_jsc_2021_06_004 crossref_primary_10_1016_j_aim_2022_108713 crossref_primary_10_1007_s13366_024_00769_4 |
| Cites_doi | 10.1142/S0218196711006674 10.1007/s00454-016-9839-6 10.1007/978-1-84996-299-5 10.1016/j.aam.2011.11.004 10.1007/s00037-012-0053-5 10.24033/bsmf.1848 10.1016/0304-3975(81)90064-5 10.1080/00927870902828793 10.1090/conm/377/06998 10.2307/1971361 10.1007/s00037-013-0077-5 10.1090/S0025-5718-1995-1297471-4 10.1016/S0747-7171(88)80003-8 10.1090/conm/495/09689 10.1090/S0002-9939-07-09005-3 10.1016/j.jsc.2005.11.006 10.1007/0-306-48658-X_6 10.1007/BF01200407 10.1137/09075024X 10.1016/S0168-0072(01)00055-0 10.1142/S0218196708004779 10.2307/1990996 10.1080/00927872.2013.766827 10.1016/j.aim.2016.12.004 10.4171/RMI/561 10.1090/cbms/097 10.1090/gsm/161 10.1016/j.jsc.2006.02.004 10.1007/s00029-017-0322-x 10.1007/978-3-0346-0048-4 |
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| Keywords | Min-plus algebra 68W30 Nullstellensatz Tropical algebra 52C99 14T05 52C99 Contents Tropical algebra Mathematics Subject Classification 14T05 |
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| Title | Tropical Effective Primary and Dual Nullstellensätze |
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