Algebraic Stories — The $k$-rank of monomials

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.Let $k\geq3$, $d\geq1$ and $a,b\geq0$ satisfy $a+b=kd$. Define $s,t\in\{0,\ldots,k-1\}$ by $a\equiv s\pmod k$ and $b\equiv t\pmod k$.

Statement recorded in the supplied source.

The bound $\operatorname{rk}_k(x^ay^b)\leq\max(s,t)+1$ is given in the source. Is it always an equality? \[\operatorname{rk}_k(x^ay^b)=\max(s,t)+1.\] [S2]

Short English statement

Is the known upper bound for binary monomials, determined by their exponents modulo $k$, always exact?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500006 (AMR-014-0006). 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), §1.3, Problem D. https://arxiv.org/abs/1801.01692.