Algebraic Stories — Hilbert series of other classes of 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 $d,n,r\geq1$ and let $\mu=(\mu_1,\ldots,\mu_q)$ be a partition of $d$ into positive parts. For $i=1,\ldots,r$ choose generic linear forms $\ell_{i,1},\ldots,\ell_{i,q}$, distinct within each product, and set \[I=\left(\prod_{j=1}^q\ell_{1,j}^{\mu_j},\ldots,\prod_{j=1}^q\ell_{r,j}^{\mu_j}\right). \] This is a generic $\mu$-power ideal.

Statement recorded in the supplied source.

For every $\mu\neq(d)$, is the Hilbert function of $S/I$ the coefficient sequence of \[\left[\frac{(1-t^d)^r}{(1-t)^n}\right]_+?\] [S2]

Short English statement

Do products of powers of distinct generic linear forms have the predicted generic-forms Hilbert function?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500010 (AMR-014-0010). 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), §2.2, Problem F. https://arxiv.org/abs/1801.01692.