Binary forms with three different relative ranks

Reznick, Bruce; Tokcan, Neriman

doi:10.1090/proc/13666

Binary forms with three different relative ranks
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by Bruce Reznick and Neriman Tokcan PDF

Proc. Amer. Math. Soc. 145 (2017), 5169-5177 Request permission

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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