Algebraic Stories — Generic $k$-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.

Work over $\mathbb C$. For $n\geq1$, let $S=\mathbb C[x_1,\ldots,x_n]$, with $S_e$ its vector space of homogeneous polynomials of total degree $e$. For $k,d\geq1$ and $f\in S_{kd}$, define \[\operatorname{rk}_k(f)=\min\left\{s\geq0:f=\sum_{j=1}^{s}g_j^k,\quad g_j\in S_d\right\}. \] A general form means one whose coefficient vector lies in a suitable nonempty Zariski open subset of $S_{kd}$. Write $\operatorname{rk}_k^\circ(kd,n)$ for the value on general forms.

Statement recorded in the supplied source.

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

Short English statement

Determine how many powers of degree-$d$ complex forms are needed for a general degree-$kd$ form in any number of variables.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500001 (AMR-014-0001). 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), §1.1, Problem A. https://arxiv.org/abs/1801.01692.