Algebraic Stories — Degree of the Waring map

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 $k\geq2$, $d\geq1$. Call $(k,d)$ perfect when $s=(kd+1)/(d+1)$ is an integer. For a perfect pair define \[W_{k,d}:S_d^s\longrightarrow S_{kd},\qquad(g_1,\ldots,g_s)\longmapsto\sum_{j=1}^sg_j^k. \] Let $\widetilde W_{k,d}:S_d^s/\mathfrak S_s\to S_{kd}$ be the induced map, where the symmetric group permutes the entries. Its degree is the number of unordered tuples in the fibre over a general form. This definition uses the permutation quotient specified in the source.

Statement recorded in the supplied source.

Calculate $\deg(\widetilde W_{k,d})$ for every perfect pair $(k,d)$. [S2]

Short English statement

Count the decompositions of a general binary form for every perfect pair, with the tuple counted up to permutation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500007 (AMR-014-0007). 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), §1.4, Definition 1.11 and Problem E. https://arxiv.org/abs/1801.01692.