For every field k of characteristic zero and all positive integers n,r,d, let S=k[x_1,…,x_n] and let f_1,…,f_r∈ S_d be generic homogeneous forms of degree d. Then operatornameHilb_S/(f_1,…,f_r)(t)=[frac(1-t^d)^r(1-t)^n]_+, where, if frac(1-t^d)^r(1-t)^n=∑_i≥ 0c_i t^i and q=mini:c_i≤ 0, the truncation is [frac(1-t^d)^r(1-t)^n]_+=∑_i=0^q-1c_i t^i; if no such q exists, the full power series is retained.

Fröberg’s conjecture

For every field $k$ of characteristic zero and all positive integers $n,r,d$ , let $S=k[x_1,\ldots,x_n]$ and let $f_1,\ldots,f_r\in S_d$ be generic homogeneous forms of degree $d$ . Then $\operatorname{Hilb}_{S/(f_1,\ldots,f_r)}(t)=\left[\frac{(1-t^d)^r}{(1-t)^n}\right]_+$ , where, if $\frac{(1-t^d)^r}{(1-t)^n}=\sum_{i\geq 0}c_i t^i$ and $q=\min\{i:c_i\leq 0\}$ , the truncation is $\left[\frac{(1-t^d)^r}{(1-t)^n}\right]_+=\sum_{i=0}^{q-1}c_i t^i$ ; if no such $q$ exists, the full power series is retained.

References

Primary source

arXiv

Additional references

Fröberg's Conjecture for Quintics and Septics in Four Variables — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper reports major progress for quintics and septics in four variables, but the full conjecture remains open.

Fröberg’s conjecture, posed by Ralf Fröberg in 1985, predicts the Hilbert series for quotients by generic homogeneous polynomials. The unrestricted conjecture is not settled.

Known results

Fröberg, 1985: the conjecture holds in two variables.
Anick, 1986: it holds in three variables.
Stanley and Watanabe: it holds for $r=n+1$ in characteristic $0$ .
Equal-degree cases and further conditional ranges were established in partial work through 2015 and later.

August 25, 2026 four-variable advance

A paper titled Fröberg's Conjecture for Quintics and Septics in Four Variables reports the conjectured series for every generator count in the quintic and septic cases, using endpoint-rank certificates and modular nonvanishing. This extends the known range but does not settle the unrestricted conjecture; the reported result remains unverified here.

Current status (as of August 2026): The conjecture is claimed for all generator counts in the four-variable quintic and septic cases, while the unrestricted conjecture remains open and that advance is unverified.

Sources

No solutions have been posted yet.