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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 · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

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 November 27, 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. 27 Nov 2020.

Vancouver:

Sen A. Module Grobner Bases Over Fields With Valuation. [Internet] [Masters thesis]. Indian Institute of Science; 2017. [cited 2020 Nov 27]. 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

2. Mbirika, Abukuse, III. Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties.

Degree: PhD, Mathematics, 2010, University of Iowa

URL: https://ir.uiowa.edu/etd/708

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties.
Let *R* be the polynomial ring Ζ[x_{1},…,x_{n}]. We present two different ideals in *R*. Both are parametrized by a Hessenberg function *h*, namely a nondecreasing function that satisfies *h(i) ≥ i* for all *i*. The first ideal, which we call <em>I_{h}</em>, is generated by modified elementary symmetric functions. The ideal *I_h* generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for <em>I_{h}</em> is redundant. We give a minimal generating set for this ideal. The second ideal, which we call <em>J_{h}</em>, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for <em>J_{h}</em>, we identify a basis for the quotient <em>R/J_{h}</em>.
We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When <em>h>h'</em>, we have <em>I_{h} ⊂ I_{h'}</em> and <em>J_{h} ⊂ J_{h'}</em>. We prove that <em>I_{h}</em> equals <em>J_{h}</em> when *h* is maximal. Since <em>I_{h}</em> is the ideal generated by the elementary symmetric functions when *h* is maximal, the generating set for <em>J_{h}</em> forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient <em>R/J_{h}</em> gives another description of the cohomology ring of the full flag variety.
The generators of the ring <em>R/J_{h}</em> are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator *X* and a Hessenberg function *h*. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as *h* varies, the quotient <em>R/J_{h}</em> may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.
*Advisors/Committee Members: Tymoczko, Julianna, 1975- (supervisor).*

Subjects/Keywords: algebraic geometry; cohomology; combinatorics; commutative ring theory; Grobner basis; symmetric functions; Mathematics

…combinatorial representation *theory* program where
I learned the fun of combinatorics. Thanks also to… …ideal form a Gr¨obner *basis*, which is a useful property. Using the Gr¨obner *basis* for
Jh , we… …identify a *basis* for the quotient R/Jh .
We introduce a partial ordering on the Hessenberg… …for Jh forms a Gr¨obner *basis* for the elementary symmetric functions.
Moreover, the quotient… …with the Garsia-Procesi *basis* B(µ) . . . . . .
2.4 Barriers to applying this…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Mbirika, Abukuse, I. (2010). Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties. (Doctoral Dissertation). University of Iowa. Retrieved from https://ir.uiowa.edu/etd/708

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

Mbirika, Abukuse, III. “Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties.” 2010. Doctoral Dissertation, University of Iowa. Accessed November 27, 2020. https://ir.uiowa.edu/etd/708.

MLA Handbook (7^{th} Edition):

Mbirika, Abukuse, III. “Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties.” 2010. Web. 27 Nov 2020.

Vancouver:

Mbirika, Abukuse I. Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties. [Internet] [Doctoral dissertation]. University of Iowa; 2010. [cited 2020 Nov 27]. Available from: https://ir.uiowa.edu/etd/708.

Council of Science Editors:

Mbirika, Abukuse I. Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties. [Doctoral Dissertation]. University of Iowa; 2010. Available from: https://ir.uiowa.edu/etd/708