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.Let $n,r,k\geq1$ and $d>1$. Choose generic $g_1,\ldots,g_r\in S_d$, and independently choose generic $f_1,\ldots,f_r\in S_{dk}$.

Statement recorded in the supplied source.

Is \[\operatorname{Hilb}_{S/(g_1^k,\ldots,g_r^k)}(t)=\operatorname{Hilb}_{S/(f_1,\ldots,f_r)}(t) \] for all these parameters? [S2]

Short English statement

Does taking powers of generic forms of degree at least two preserve the Hilbert series of fully generic generators of the resulting degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500011 (AMR-014-0011). 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, Conjecture 2.4. https://arxiv.org/abs/1801.01692.