Ficarra’s conjecture for monomial ideals with linear powers

Establish that for every monomial ideal I ⊂ S = K[x1, …, xn] with linear powers, the v-number of the kth power satisfies v(I^k) = a(I)k − 1 for all integers k ≥ 1, where a(I) is the initial degree of I.

Background

Ficarra’s conjecture originates from the observation that if a monomial ideal I has linear powers, then reg(Ik) = a(Ik) = a(I)k for all k ≥ 1, suggesting that v(Ik) might attain the minimal possible value a(I)k − 1. The conjecture has been verified for several classes of monomial ideals (e.g., those with depth(S/I) = 0, edge ideals with linear powers, polymatroidal ideals, and Hibi ideals), but remains open in general.

The present paper generalizes this conjecture to all graded ideals by proposing v(Ik) = a(I)k − c(I), where c(I) depends on the associated primes of I, and proves the generalized statement for multiple families. The original monomial version thus appears as a special case when all associated primes are generated by linear forms (so c(I) = 1).

References

Ficarra conjectured [10, Conjecture 2.6] that if I c S is a monomial ideal with linear powers, then v(Ik) = a(I)k - 1.

— Asymptotic behaviour and stability index of v-numbers of graded ideals  (2402.16583 - Biswas et al., 2024) in Section 3, paragraph preceding Conjecture 3.5

In , Vu proved that the local $v$-number of powers of the cover ideal of a bipartite graph is linear in $t$ for all $t\ge1$, and asked whether the global $v$-number is also linear in $t$ for all $t\ge1.

— V-numbers of powers of cover ideals of unimodular hypergraphs  (2608.18406 - Hang et al., 19 Aug 2026) in Section 1, Introduction