Algebraic Stories — Hilbert series of generic power ideals.

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 $S=\mathbb C[x_1,\ldots,x_n]$ with its usual total-degree grading. For a homogeneous quotient $R$, define \[\operatorname{Hilb}_R(t)=\sum_{e\geq0}(\dim_{\mathbb C}R_e)t^e. \] A generic tuple of forms has its coefficient vector in a suitable nonempty Zariski open subset of the product of the prescribed homogeneous pieces.For a series $A(t)=\sum_{e\geq0}a_et^e$, use $[A(t)]_+$ for its truncation at the first non-positive coefficient: if $e_0=\min\{e:a_e\leq0\}$, retain the terms with $e<e_0$ and replace all later terms by zero; if no such $e_0$ exists, retain the entire series.Let $n,r,d\geq1$, choose generic linear forms $\ell_1,\ldots,\ell_r\in S_1$, and put $R=S/(\ell_1^d,\ldots,\ell_r^d)$.

Statement recorded in the supplied source.

Outside the listed exceptional parameters \[(n,r)\in\{(3,7),(3,8),(4,9),(5,14)\}\quad\text{or}\quad r\in\{n+2,n+3\}, \] is \[\operatorname{Hilb}_R(t)=\left[\frac{(1-t^d)^r}{(1-t)^n}\right]_+? \] The source lists the four fixed pairs as exceptions and $r=n+2,n+3$ as possible further exceptions; it does not assert failure for every degree in those parameter ranges. [S2]

Short English statement

Does the generic-forms Hilbert series also describe powers of generic linear forms, apart from the stated exceptional or possibly exceptional ranges?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500009 (AMR-014-0009). 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), §2.1, Conjecture 2.3. https://arxiv.org/abs/1801.01692.