Eisenbud–Huneke–Ulrich stabilization conjecture for powers of partially linearly resolved ideals

Prove that for every integer p≥1, if an m-primary homogeneous ideal I in C[x_0,…,x_n], generated in degree d, has a minimal free resolution that is linear for p steps, then I^t=m^{td} for every t≥⌈n/p⌉.

Background

The paper studies when powers of an m-primary ideal I generated in degree d become equal to the corresponding powers of the homogeneous maximal ideal m. For linearly presented ideals, the Eisenbud–Ulrich conjecture predicts equality beginning at t=n, and the paper proves that case.

The Eisenbud–Huneke–Ulrich conjecture generalizes this prediction by replacing a fully linear presentation with the weaker condition that the resolution is linear for p steps. The conjectured stabilization threshold is ⌈n/p⌉. The paper establishes bounds in a substantial range, but its stated theorem gives the threshold s(n,p), which is strictly larger than the conjectured value for some pairs (n,p), including p=2,n=6 and several higher-dimensional cases.

References

Eisenbud, Huneke, and Ulrich conjectured in *{Conjecture~1.4} the following extension of the Eisenbud--Ulrich conjecture. For $p \ge 1$, if $I$ is an $m$-primary ideal such that its resolution is linear for $p$ steps, then $It = m{td} \quad \text{for all }t \ge \left\lceil \frac{n}{p}\right\rceil.$

— The Eisenbud-Huneke-Ulrich conjecture and bounds on minimal generators  (2609.12302 - Yang, 11 Sep 2026) in Section 1, subsection “Stabilization of powers,” Conjecture 1 (Conjecture~\ref{conj:EHU})