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(I^k) = a(I^k) = a(I)k for all k ≥ 1, suggesting that v(I^k) 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(I^k) = 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).