Relative waring rank of binary forms

Tokcan, Neriman

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https://hdl.handle.net/2142/98327 Copy

Description

Title

Relative waring rank of binary forms

Author(s)

Tokcan, Neriman

Issue Date

2017-07-05

Director of Research (if dissertation) or Advisor (if thesis)

Reznick, Bruce

Doctoral Committee Chair(s)

Bergvelt, Maarten

Committee Member(s)

• Katz, Sheldon
• Nevins, Thomas

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2017-08

Type of Resource

text

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http://hdl.handle.net/2142/98327

Copyright and License Information

Copyright 2017 Neriman Tokcan

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