Gröbner bases over fields with valuations
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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.References
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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
- MR Author ID: 607134
- Email: D.Maclagan@warwick.ac.uk
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 467-483
- MSC (2010): Primary 13P10; Secondary 14T05
- DOI: https://doi.org/10.1090/mcom/3321
- MathSciNet review: 3854067