Open Access Theses and Dissertations

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You searched for +publisher:"University of Illinois – Chicago" +contributor:("Hampton, Marshall"). Showing records 1 – 3 of 3 total matches.

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University of Illinois – Chicago

1. Bliss, Nathan R. Computing Series Expansions of Algebraic Space Curves.

Degree: 2018, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/22682 

We work towards a series-based computational approach for polynomial systems having positive-dimensional solution sets. The tropical variety gives information on the exponents of the leading terms of the series; we provide insight into when the purely polyhedral and more easily computed tropical prevariety is sufficient. When it is not sufficient and hidden cones exist, we give an alternative to known symbolic algorithms based on polyhedral end games. We develop an effective method to apply the Gauss-Newton algorithm over power or Laurent series, using linearization and a lower triangular echelon form; we can thus extend the information obtained tropically with quadratic convergence. We also characterize when tropical methods can be avoided entirely. Finally we give applications to several problems in view of extending current approaches to homotopy continuation to allow for starting from singular solutions. We also provide a result related to the Backelin component of the cyclic-16 roots polynomial system. Advisors/Committee Members: Verschelde, Jan (advisor), Awanou, Gerard (committee member), Hampton, Marshall (committee member), Reyzin, Lev (committee member), Tucker, Kevin (committee member), Verschelde, Jan (chair).

Subjects/Keywords: computational algebraic geometry; puiseux series; gauss-newton algorithm; tropical geometry; polynomial systems; homotopy continuation

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

APA (6th Edition):

Bliss, N. R. (2018). Computing Series Expansions of Algebraic Space Curves. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/22682

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Bliss, Nathan R. “Computing Series Expansions of Algebraic Space Curves.” 2018. Thesis, University of Illinois – Chicago. Accessed January 19, 2021. http://hdl.handle.net/10027/22682.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Bliss, Nathan R. “Computing Series Expansions of Algebraic Space Curves.” 2018. Web. 19 Jan 2021.

Vancouver:

Bliss NR. Computing Series Expansions of Algebraic Space Curves. [Internet] [Thesis]. University of Illinois – Chicago; 2018. [cited 2021 Jan 19]. Available from: http://hdl.handle.net/10027/22682.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bliss NR. Computing Series Expansions of Algebraic Space Curves. [Thesis]. University of Illinois – Chicago; 2018. Available from: http://hdl.handle.net/10027/22682

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Illinois – Chicago

2. Sommars, Jeffrey C. Algorithms and Implementations in Computational Algebraic Geometry.

Degree: 2018, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/22687 

In this thesis, we explore several areas of computational algebraic geometry, and develop new algorithms and software in each. We are generally interested in solving polynomial systems and applications that require solving polynomial systems. Advisors/Committee Members: Verschelde, Jan (advisor), Awanou, Gerard (committee member), Hampton, Marshall (committee member), Reyzin, Lev (committee member), Tucker, Kevin (committee member), Verschelde, Jan (chair).

Subjects/Keywords: Tropical geometry; computational algebraic geometry

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

APA (6th Edition):

Sommars, J. C. (2018). Algorithms and Implementations in Computational Algebraic Geometry. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/22687

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Sommars, Jeffrey C. “Algorithms and Implementations in Computational Algebraic Geometry.” 2018. Thesis, University of Illinois – Chicago. Accessed January 19, 2021. http://hdl.handle.net/10027/22687.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Sommars, Jeffrey C. “Algorithms and Implementations in Computational Algebraic Geometry.” 2018. Web. 19 Jan 2021.

Vancouver:

Sommars JC. Algorithms and Implementations in Computational Algebraic Geometry. [Internet] [Thesis]. University of Illinois – Chicago; 2018. [cited 2021 Jan 19]. Available from: http://hdl.handle.net/10027/22687.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Sommars JC. Algorithms and Implementations in Computational Algebraic Geometry. [Thesis]. University of Illinois – Chicago; 2018. Available from: http://hdl.handle.net/10027/22687

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

3. Adrovic, Danko. Solving Polynomial Systems With Tropical Methods.

Degree: 2013, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/9735 

In this thesis, we develop a polyhedral method to solve polynomial systems. We are primarily interested in obtaining the Puiseux series representations of positive dimensional solution sets for square polynomial systems and systems, which consist of more equations than unknowns. By developing our polyhedral method, we aim to generalize polyhedral homotopies. Our polyhedral method can be seen as the symbolic-numeric version of the fundamental theorem of tropical algebraic geometry. We illustrate our polyhedral method on the cyclic n-roots problems and offer a tropical perspective on the lemma of Backelin. Advisors/Committee Members: Verschelde, Jan (advisor), Culler, Marc (committee member), Dumas, David (committee member), Greenblatt, Michael (committee member), Hampton, Marshall (committee member).

Subjects/Keywords: Newton-Puiseux method; polyhedral homotopies; Puiseux series; tropism; initial forms; unimodular coordinate transformations; cyclic n-roots problem

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

APA (6th Edition):

Adrovic, D. (2013). Solving Polynomial Systems With Tropical Methods. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/9735

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Adrovic, Danko. “Solving Polynomial Systems With Tropical Methods.” 2013. Thesis, University of Illinois – Chicago. Accessed January 19, 2021. http://hdl.handle.net/10027/9735.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Adrovic, Danko. “Solving Polynomial Systems With Tropical Methods.” 2013. Web. 19 Jan 2021.

Vancouver:

Adrovic D. Solving Polynomial Systems With Tropical Methods. [Internet] [Thesis]. University of Illinois – Chicago; 2013. [cited 2021 Jan 19]. Available from: http://hdl.handle.net/10027/9735.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Adrovic D. Solving Polynomial Systems With Tropical Methods. [Thesis]. University of Illinois – Chicago; 2013. Available from: http://hdl.handle.net/10027/9735

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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