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.
|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|
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