Cited By
View all- Hampe SJoswig M(2018)Tropical Computations in polymakeAlgorithmic and Experimental Methods in Algebra, Geometry, and Number Theory10.1007/978-3-319-70566-8_14(361-385)Online publication date: 23-Mar-2018
Computing tropical varieties over fields with valuation using classical standard basis techniques
Author: 
Yue RenAuthors Info & Claims
Pages 127 - 129
Abstract
In this presentation, we discuss the application of classical Gröbner basis (or standard basis) techniques to algebraic problems in polynomial rings over fields with non-trivial valuation arising in tropical geometry.
The key problem we will study is the computation of a tropical variety, given its ideal in the polynomial ring. All algorithms in this presentation have been implemented in the computer algebra system Singular and are available as part of the official distribution.
References
[1]
Elizabeth Baldwin, Paul Klemperer. Tropical geometry to analyze demand. preliminary draft available at http://www.nuff.ox.ac.uk/users/klemperer/tropical.pdf, 2015.
[2]
Tristam Bogart, Anders N. Jensen, David Speyer, Bernd Sturmfels, Rekha R. Thomas. Computing tropical varieties. Journal of Symbolic Computation, 2007.
[3]
Andrew J. Chan, Diane Maclagan. Gröbner bases over fields with valuations. arXiv:1303.0729, 2013.
[4]
Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, Hans Schönemann. Singular 4-0-2 --- A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2015.
[5]
Ambedkar Dukkipati, Aritra Sen. Gröbner Basis Theory for Modules over Polynomial Rings over Fields with Valuation. arXiv:1402.6675, 2014.
[6]
Anders N. Jensen. Gfan, a software system for Gröbner fans and tropical varieties. Available at http://home.imf.au.dk/jensen/software/gfan/gfan.html.
[7]
Anders N. Jensen, Frank Seelisch, Yue Ren. gfanlib.so. A Singular 4-0-2 interface to Gfanlib for convex geometry, 2015.
[8]
Diane Maclagan, Bernd Sturmfels. Introduction to Tropical Geometry. Graduate Studies in Mathematics, American Mathematical Society, 2014.
[9]
Lior Pachter, Bernd Sturmfels. Tropical geometry of statistical models. Proceedings of the National Academy of Sciences of the United States of America, 2004.
[10]
Tristan Vaccon. Matrix-F5 algorithms and tropical Gröbner bases computation. arXiv:1402.6675, 2014.
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Copyright © 2016 Author.
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Association for Computing Machinery
New York, NY, United States
Publication History
Published: 17 February 2016
Published in SIGSAM-CCA Volume 49, Issue 4
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Cited By
View all- Hampe SJoswig M(2018)Tropical Computations in polymakeAlgorithmic and Experimental Methods in Algebra, Geometry, and Number Theory10.1007/978-3-319-70566-8_14(361-385)Online publication date: 23-Mar-2018
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