Typical Real Ranks of Binary Forms

Abstract

We prove a conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree $d$ take all integer values between $\lfloor \frac{d+2}{2}\rfloor$ and $d$. That is we show that for all $d$ and all $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a bivariate form $f$ such that $f$ can be written as a linear combination of $m$ $d$-th powers of real linear forms and no fewer, and additionally all forms in an open neighborhood of $f$ also possess this property. Equivalently we show that for all $d$ and any $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a symmetric real bivariate tensor $t$ of order $d$ such that $t$ can be written as a linear combination of $m$ symmetric real tensors of rank 1 and no fewer, and additionally all tensors in an open neighborhood of $t$ also possess this property.