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 $k,d,n\geq1$ and set $T=\mathbb C[x_0,\ldots,x_n]$. For $f\in T_{kd}$, its $k$-rank is the least $s\geq0$ such that $f=g_1^k+\cdots+g_s^k$ with homogeneous $g_j\in T_d$. Using the $n+1$-variable convention of Definition 1.8, put \[\operatorname{rk}^{\max}_k(kd,n)=\max_{f\in T_{kd}}\operatorname{rk}_k(f). \] Equivalently, it is the smallest uniform bound on the number of summands required for every degree-$kd$ form in $n+1$ variables.

Statement recorded in the supplied source.

Determine $\operatorname{rk}^{\max}_k(kd,n)$ for every triple of positive integers $(k,d,n)$. [S2]

Short English statement

Determine the largest number of powers required by a complex form with given degree and number of variables.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500003 (AMR-014-0003). 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, Definition 1.8 and Problem B. https://arxiv.org/abs/1801.01692.