dc.contributor.author Romero Barbosa, Julian dc.date.accessioned 2016-09-23 18:27:42 (GMT) dc.date.available 2016-09-23 18:27:42 (GMT) dc.date.issued 2016-09-23 dc.date.submitted 2016-09-13 dc.identifier.uri http://hdl.handle.net/10012/10897 dc.description.abstract Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \textit{stable set} of a given size, finding the \textit{chromatic number} of a graph or more generally, determining the feasibility of an \textit{Integer Programming problem} are classical examples of this. In this thesis we study a powerful tool from Algebraic Geometry, called \textit{Hilbert's Nullstellensatz}. It characterizes the \textit{infeasibility} of a system of polynomial equations by the \textit{feasibility} of a possibly very large system of \textit{linear equations}. The solutions to this linear system provide \textit{certificates} for the infeasibility of the polynomial system, called \textit{Nullstellensatz Certificates}. en In this thesis we focus on the study of Nullstellensatz Certificates for the existence of \textit{proper colorings} of graphs. We use basic ideas from \textit{duality theory} to determine various properties of the Nullstellensatz Certificates. We give new proofs to several known results in the current literature and present some new results that shed some light on the relationship between the sparsity of a graph and the \textit{size} of the Nullstellensatz Certificates for \textit{$k$-colorability}. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Algebraic Geometry en dc.subject Combinatorial Optimization en dc.subject Hilbert's Nullstellensatz en dc.subject Graph Coloring en dc.subject Graph Theory en dc.title Applied Hilbert's Nullstellensatz for Combinatorial Problems en dc.type Master Thesis en dc.pending false uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree.discipline Combinatorics and Optimization en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Master of Mathematics en uws.contributor.advisor Tunçel, Levent uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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