Fröberg conjecture on the Hilbert functions of a set of 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.

Work over $\mathbb C$, as in the source's convention. Fix $n,r\geq1$ and degrees $d_1,\ldots,d_r\geq1$. Let $f_i\in\mathbb C[x_1,\ldots,x_n]$ be generic homogeneous forms of degrees $d_i$: the coefficients are chosen in a suitable nonempty Zariski open subset of the product of coefficient spaces. Set $R=\mathbb C[x_1,\ldots,x_n]/(f_1,\ldots,f_r)$ and $H_R(t)=\sum_{j\geq0}\dim_{\mathbb C}(R_j)t^j$, where $R_j$ is the homogeneous degree-$j$ part. For a series $A(t)=\sum_{j\geq0}a_jt^j$, define $[A(t)]_+=\sum_{j\geq0}b_jt^j$, where $b_j=a_j$ if $a_i\geq0$ for every $i\leq j$, and $b_j=0$ otherwise. Thus truncation starts at the first negative coefficient, if there is one.

Statement recorded in the supplied source.

For every such choice of $n,r,d_1,\ldots,d_r$, do generic forms satisfy \[H_R(t)=\left[\frac{\prod_{i=1}^r(1-t^{d_i})}{(1-t)^n}\right]_+?\] [S2]

Short English statement

Does the quotient by generic forms have the Hilbert series predicted by the product formula, truncated from its first negative coefficient onward?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200006 (AMR-071-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ralf Fröberg and Samuel Lundqvist, Extremal Hilbert series (2017). Sections 1–2, Conjecture 1, pp. 1–3. https://arxiv.org/pdf/1711.01232v1.