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Mathematics of Computation

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Gröbner bases over fields with valuations


Authors: Andrew J. Chan and Diane Maclagan
Journal: Math. Comp. 88 (2019), 467-483
MSC (2010): Primary 13P10; Secondary 14T05
DOI: https://doi.org/10.1090/mcom/3321
Published electronically: April 6, 2018
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Abstract: Let $ K$ be a field with a valuation and let $ S$ be the polynomial ring $ S:= K[x_1, \dots , x_n]$. 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 =\mathbb{Q}$ with the $ p$-adic valuation the algorithms have been implemented in a Macaulay 2 package.


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Additional Information

Andrew J. Chan
Affiliation: Mathematics Institute University of Warwick Coventry, CV4 7AL United Kingdom
Email: andrew.john.chan@gmail.com

Diane Maclagan
Affiliation: Mathematics Institute University of Warwick Coventry, CV4 7AL United Kingdom
Email: D.Maclagan@warwick.ac.uk

DOI: https://doi.org/10.1090/mcom/3321
Received by editor(s): October 28, 2016
Received by editor(s) in revised form: July 29, 2017
Published electronically: April 6, 2018
Additional Notes: The second author was partially supported by EPSRC grant EP/I008071/1.
Article copyright: © Copyright 2018 American Mathematical Society

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