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Indian Institute of Science

1.
Sen, Aritra.
Module *Grobner* Bases Over Fields With Valuation.

Degree: MSc Engg, Faculty of Engineering, 2017, Indian Institute of Science

URL: http://etd.iisc.ac.in/handle/2005/2644

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension.
In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector.
We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.
*Advisors/Committee Members: Dukkipati, Ambedkar (advisor).*

Subjects/Keywords: Grobner Basis; Tropical Algebraic Geometry; Grobner Basis Theory; Hilbert Polynomials; Syzygies; Free Resolutions; Computational Geometry; Grobner Basis Computation; Algebraic Geometry; Tropical Geometry; Grobner Bases; Mathematics

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APA (6^{th} Edition):

Sen, A. (2017). Module Grobner Bases Over Fields With Valuation. (Masters Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/2644

Chicago Manual of Style (16^{th} Edition):

Sen, Aritra. “Module Grobner Bases Over Fields With Valuation.” 2017. Masters Thesis, Indian Institute of Science. Accessed December 02, 2020. http://etd.iisc.ac.in/handle/2005/2644.

MLA Handbook (7^{th} Edition):

Sen, Aritra. “Module Grobner Bases Over Fields With Valuation.” 2017. Web. 02 Dec 2020.

Vancouver:

Sen A. Module Grobner Bases Over Fields With Valuation. [Internet] [Masters thesis]. Indian Institute of Science; 2017. [cited 2020 Dec 02]. Available from: http://etd.iisc.ac.in/handle/2005/2644.

Council of Science Editors:

Sen A. Module Grobner Bases Over Fields With Valuation. [Masters Thesis]. Indian Institute of Science; 2017. Available from: http://etd.iisc.ac.in/handle/2005/2644