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 Title: Relative waring rank of binary forms Author(s): Tokcan, Neriman Director of Research: Reznick, Bruce Doctoral Committee Chair(s): Bergvelt, Maarten Doctoral Committee Member(s): Katz, Sheldon; Nevins, Thomas Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Waring rank, Real rank, Binary forms, Sums of powers, Sylvester, Tensor decompositions Abstract: Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \cc$. The {\it $K$-rank of $f$} is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. We also study the relation between the relative rank and the algebraic properties of the underlying field. In particular, we show that $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether $-1$ is a sum of two squares in $K.$ We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We also give the rank of quartic and quintic binary forms based on their factorization over $\cc.$ We investigate the structure of binary forms with unique $\mathbb{C}$-minimal representation. Issue Date: 2017-07-05 Type: Thesis URI: http://hdl.handle.net/2142/98327 Rights Information: Copyright 2017 Neriman Tokcan Date Available in IDEALS: 2017-09-29 Date Deposited: 2017-08
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