Gröbner fans of x-homogeneous ideals in R〚t〛[x]. (November 2017)

• Record Type: Journal Article
• Authors: Markwig, Thomas; Ren, Yue
• Is Part Of: Journal of symbolic computation. Volume 83(2017)
• Page Start: 315
• Page End: 341
• Publication Date: 2017-11
• Subjects: 13P10 -- 13F25 -- 16W60 -- 12J25 -- 16W60; Gröbner fan -- Standard basis over base rings -- Local standard fan -- Tropical variety; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.11.016
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We generalise the notion of a Gröbner fan to ideals in R 〚 t 〛 [ x 1, …, x n ] for certain classes of coefficient rings R and give a constructive proof that the Gröbner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gröbner fan, implemented in the computer algebra systemSingular . The problem is motivated by the wish to compute tropical varieties over the p -adic numbers.