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Module Grobner Bases Over Fields With Valuation

Electronic Theses of Indian Institute of Science

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Title Module Grobner Bases Over Fields With Valuation
 
Creator Sen, Aritra
 
Subject 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
 
Description 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.
 
Contributor Dukkipati, Ambedkar
 
Date 2017-07-12T10:47:34Z
2017-07-12T10:47:34Z
2017-07-12
2015-01
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2644
http://etd.ncsi.iisc.ernet.in/abstracts/3448/G26717-Abs.pdf
 
Language en_US
 
Relation G26717