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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.
| 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 |
| Source Type: | DISSERTATION |
| Subjects: | Mathematics, Theoretical Mathematics |
| Keywords: | Algebraic Geometry, Buchberger, Gröbner, Khovanskii, Macalay2, Tropical Variety |
| Publication Number: | 27834094 |
| ISBN: | 9798672197098 |
