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Gröbner bases over fields with valuations
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Chan, Andrew J. and Maclagan, Diane (2019) Gröbner bases over fields with valuations. Mathematics of Computation, 88 . pp. 467-483. doi:10.1090/mcom/3321 ISSN 0025-5718.
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Official URL: https://doi.org/10.1090/mcom/3321
Abstract
Let K be a field with a valuation and let S be the polynomial ring S := K[x1; : : : ; xn]. We discuss the extension of Gröbner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case K = Q with the p-adic valuation the algorithms have been implemented in a Macaulay 2 package.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Gröbner bases, Algorithms | ||||||||
Journal or Publication Title: | Mathematics of Computation | ||||||||
Publisher: | American Mathematical Society | ||||||||
ISSN: | 0025-5718 | ||||||||
Official Date: | 2019 | ||||||||
Dates: |
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Volume: | 88 | ||||||||
Page Range: | pp. 467-483 | ||||||||
DOI: | 10.1090/mcom/3321 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 4 September 2017 | ||||||||
Date of first compliant Open Access: | 25 October 2019 | ||||||||
RIOXX Funder/Project Grant: |
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