Algebraic Stories — Ideals of generic forms

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.Fix positive integers $n,r,d_1,\ldots,d_r$, and choose generic $f_i\in S_{d_i}$.

Statement recorded in the supplied source.

Does $R=S/(f_1,\ldots,f_r)$ have the Hilbert series \[\operatorname{Hilb}_R(t)=\left[\frac{\prod_{i=1}^r(1-t^{d_i})}{(1-t)^n}\right]_+?\] [S2]

Short English statement

Does the proposed truncated product give the Hilbert series for every ideal generated by generic complex forms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500008 (AMR-014-0008). 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, Conjecture 2.2. https://arxiv.org/abs/1801.01692.