Relative waring rank of binary forms

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 x³y²) 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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Tokcan, Neriman

Contributors dc:contributor

• Reznick, Bruce
• Bergvelt, Maarten
• Katz, Sheldon
• Nevins, Thomas

• Waring rank
• Real rank
• Binary forms
• Sums of powers
• Sylvester
• Tensor decompositions

Statement dc:rights

• Copyright 2017 Neriman Tokcan

Language dc:language

en

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F98327