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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Tropical noetherity and Gröbner bases

Authors: Ya. Kazarnovskiĭ and A. G. Khovanskiĭ
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 797-811
MSC (2010): Primary 16S34
Published electronically: July 27, 2015
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Abstract: A set that is a Gröbner basis for an ideal with respect to every Gröbner ordering is called a universal Gröbner basis for that ideal. In the paper, it is proved that there exists a universal Gröbner basis in which the polynomials have controlled degrees. The main result is the theorem on the tropical Noetherity of a ring of Laurent polynomials. This theorem is close to the existence theorem for a universal basis and is needed for the tropical intersection theory in  $ (\mathbb{C}^*)^n$, which will be presented in a forthcoming paper.

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Additional Information

Ya. Kazarnovskiĭ
Affiliation: Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia

A. G. Khovanskiĭ
Affiliation: Institute for Systems Analysis, Russian Academy of Sciences, 60-letiya Oktyabrya pr. 9, Moscow 117312; Independent University of Moscow, Bolshoy Vlasyevskiǐ Pereulok 11, Moscow 119002, Russia; University Of Toronto, Canada

Keywords: Laurent polynomial, ideal, tropical basis, universal Gr\"obner basis, Seidenberg theorem
Received by editor(s): October 17, 2013
Published electronically: July 27, 2015
Additional Notes: The first author was partially supported by the grant SS-4850.2012.1; the second author was partially supported by the Canadian grant 0GP0156833
Article copyright: © Copyright 2015 American Mathematical Society

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