Algebraic Stories — Maximal $k$-rank

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $S=\mathbb C[x,y]$, and let $S_e$ denote its homogeneous degree-$e$ forms. For $f\in S_{kd}$, let $\operatorname{rk}_k(f)$ be the smallest number of terms in a decomposition $f=\sum_jg_j^k$ with $g_j\in S_d$. Define $R_{k,d}=\max_{f\in S_{kd}}\operatorname{rk}_k(f)$, the maximal $k$-rank of binary forms.

Statement recorded in the supplied source.

For all positive integers $k,d$, is $R_{k,d}=k$? In addition, for every pair of non-proportional linear forms $\ell_1,\ell_2\in S_1$, is \[\operatorname{rk}_k(\ell_1\ell_2^{kd-1})=k?\] [S2]

Short English statement

Are $k$ powers always enough for binary forms, with the stated products of linear forms attaining the maximum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500004 (AMR-014-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. Fröberg, S. Lundqvist, A. Oneto and B. Shapiro, Algebraic Stories from One and from the Other Pockets, arXiv:1801.01692 (2018), §1.2, Conjecture 1.9. https://arxiv.org/abs/1801.01692.