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Binary forms with three different relative ranks


Authors: Bruce Reznick and Neriman Tokcan
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11E76, 11P05, 12D15, 14N10
DOI: https://doi.org/10.1090/proc/13666
Published electronically: June 16, 2017
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Abstract: Suppose $ f(x,y)$ is a binary form of degree $ d$ with coefficients in a field $ K \subseteq \mathbb{C}$. The $ 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. The $ K$-rank of a form $ f$ (such as $ x^3y^2$) may depend on whether -1 is a sum of two squares in $ K$.


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Additional Information

Bruce Reznick
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: reznick@illinois.edu

Neriman Tokcan
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043
Email: tokcan2@illinois.edu

DOI: https://doi.org/10.1090/proc/13666
Keywords: Complex rank, real rank, binary forms, sums of powers, Stufe, Sylvester, tensor decompositions
Received by editor(s): August 26, 2016
Received by editor(s) in revised form: January 12, 2017
Published electronically: June 16, 2017
Additional Notes: Part of the work in this paper is taken from the doctoral dissertation of the second author, written under the direction of the first author. The first author was supported in part by Simons Collaboration Grant 280987.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2017 American Mathematical Society