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Some combinatorial results on matrices and polynomials
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Title
Some combinatorial results on matrices and polynomials
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Semonsen
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Justin
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Semonsen, Justin
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Kopparty
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Swastik
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Swastik Kopparty
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chair
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Saraf
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Shubhangi
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Shubhangi Saraf
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internal member
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Narayanan
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Bhargav
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Bhargav Narayanan
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internal member
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Srinivasan
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Srikanth
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Srikanth Srinivasan
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Advisory Committee
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Rutgers University
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School of Graduate Studies
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theses
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2020
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2020-05
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English
Abstract
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This thesis studies three problems in combinatorics that concern matrices and polyno- mials.
The first problem refines a technique by Scheinerman [24] to get an improved upper bound on the determinants of n×n 0-1 matrices with k ones in each row. Our new bound uses linear programming techniques to analyze a greedy decomposition algorithm, show how it improves on previous methods, and determine its asymptotic behavior as a function of k. We also present and analyze an improvement to this method, as well as the limitations and other possibilities of using this technique.
The second problem analyzes the supports of F2 polynomials on n variables in Ham- ming balls, and proves an optimal Schwartz-Zippel-like bound. We then use methods based on those of Kasami and Tokura [11], [12] to find and classify all tight polynomi- als for this bound. Our result is based on studying necessary conditions for ”division lemmas” for polynomials.
The third result studies sets of points in F2^n for which the sum of any F2 polynomial of degree d on those points is 0. We prove that for d ≥ 2, we have that the size of the set must at least twice the affine dimension of the set. For larger d, we can show the size of the set must be a constant amount larger than twice the affine dimension, but we conjecture that this can be improved to 2^(d+1)/(d+2) times the affine dimension of the set. We also apply these theorems to prove a bound on the weight distribution of Reed-Muller codes of high dimension.
The first problem refines a technique by Scheinerman [24] to get an improved upper bound on the determinants of n×n 0-1 matrices with k ones in each row. Our new bound uses linear programming techniques to analyze a greedy decomposition algorithm, show how it improves on previous methods, and determine its asymptotic behavior as a function of k. We also present and analyze an improvement to this method, as well as the limitations and other possibilities of using this technique.
The second problem analyzes the supports of F2 polynomials on n variables in Ham- ming balls, and proves an optimal Schwartz-Zippel-like bound. We then use methods based on those of Kasami and Tokura [11], [12] to find and classify all tight polynomi- als for this bound. Our result is based on studying necessary conditions for ”division lemmas” for polynomials.
The third result studies sets of points in F2^n for which the sum of any F2 polynomial of degree d on those points is 0. We prove that for d ≥ 2, we have that the size of the set must at least twice the affine dimension of the set. For larger d, we can show the size of the set must be a constant amount larger than twice the affine dimension, but we conjecture that this can be improved to 2^(d+1)/(d+2) times the affine dimension of the set. We also apply these theorems to prove a bound on the weight distribution of Reed-Muller codes of high dimension.
Subject
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Mathematics
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Rutgers University Electronic Theses and Dissertations
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1 online resource (viii, 48 pages)
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Ph.D.
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Includes bibliographical references
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School of Graduate Studies Electronic Theses and Dissertations
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doi:10.7282/t3-a5pa-d134
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ETD doctoral
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The author owns the copyright to this work.
RightsHolder
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Name
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Semonsen
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Justin
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Permission or license
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2020-03-28 16:17:30
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Justin Semonsen
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Rutgers University. School of Graduate Studies
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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