Ehrmann, Daniel
(2020)
Khovanskii-Gröbner Basis.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
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.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
8 June 2020 |
| Date Type: |
Publication |
| Defense Date: |
20 November 2019 |
| Approval Date: |
8 June 2020 |
| Submission Date: |
1 May 2020 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
83 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Algebra, Gröbner Basis, Gröbner Bases, Algorithm, Khovanskii Basis, Khovanskii Bases, Algebraic Geometry |
| Date Deposited: |
08 Jun 2020 16:10 |
| Last Modified: |
08 Jun 2020 16:10 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/38858 |
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