Dissertation/Thesis Abstract

Khovanskii-Gröbner Bases
by Ehrmann, Daniel James, Ph.D., University of Pittsburgh, 2020, 84; 27834094
Abstract (Summary)

In this thesis a natural generalization and further extension of Gröbner theory using Kaveh and Manon's Khovanskii basis theory is constructed. Suppose A is a finitely generated domain equipped with a valuation v with a finite Khovanskii basis. We develop algorithmic processes for computations regarding ideals in the algebra A. We introduce the notion of a Khovanskii-Gröbner basis for an ideal J in A and give an analogue of the Buchberger algorithm for it (accompanied by a Macaulay2 code). We then use Khovanskii- Gröbner bases to suggest an algorithm to solve a system of equations from A. Finally we suggest a notion of relative tropical variety for an ideal in A and sketch ideas to extend the tropical compactification theorem to this setting.

Indexing (document details)
Advisor: Kaveh, Kiumars
Commitee: Hales, Thomas, Manon, Christopher, Constantine, Gregory
School: University of Pittsburgh
Department: Dietrich School Arts and Sciences
School Location: United States -- Pennsylvania
Source: DAI-B 82/3(E), Dissertation Abstracts International
Subjects: Mathematics, Theoretical Mathematics
Keywords: Algebraic Geometry, Buchberger, Gröbner, Khovanskii, Macalay2, Tropical Variety
Publication Number: 27834094
ISBN: 9798672197098
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