Sharp lower bound for minimal generators of virtually partially linearly resolved ideals

Prove that if an m-primary homogeneous ideal I in C[x_0,…,x_n], generated in degree d, has a resolution that is virtually linear for p steps, then its number of minimal generators satisfies μ(I)≥N(n,p,d)=\binom{p+d}{p}+(n-p)\binom{p+d-1}{p}.

Background

The paper derives lower bounds for the number μ(I) of minimal generators from the equality It=m{td} and from geometric restrictions imposed by virtual linearity. The proposed bound N(n,p,d) is designed to vary with the number p of virtually linear steps, unlike simpler bounds obtained from a single power equality.

The conjecture is proved in the paper when p=1 or d=2. The final remark explicitly states that the conjecture has not been established in full generality, so the cases with general p and d remain unresolved.

References

This motivates the following conjecture. Set $N(n,p,d) \coloneqq \binom{p+d}{p} + (n-p)\binom{p+d-1}{p}$. If $I$ is $m$-primary and its resolution is virtually linear for $p$ steps, then $\mu(I) \ge N(n,p,d)$.

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