## Binary forms with three different relative ranks

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- by Bruce Reznick and Neriman Tokcan
- Proc. Amer. Math. Soc.
**145**(2017), 5169-5177 - 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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## Bibliographic Information

**Bruce Reznick**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 147525
- 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
- MR Author ID: 1204531
- Email: tokcan2@illinois.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 5169-5177 - MSC (2010): Primary 11E76, 11P05, 12D15, 14N10
- DOI: https://doi.org/10.1090/proc/13666
- MathSciNet review: 3717946