Generalized Ficarra Conjecture for graded ideals with linear powers

Establish that for every graded ideal I ⊂ S = K[x1, …, xn] with linear powers, the v-number of the kth power satisfies v(I^k) = a(I)k − c(I) for all integers k ≥ 1, where a(I) is the initial degree of I and c(I) = max{a(p) : p ∈ Ass(I)} is the maximum initial degree among the associated primes of I.

Background

The paper studies the asymptotic behavior of the v-number v(I^k) of powers of graded ideals I in a standard graded polynomial ring S. Earlier work of Ficarra and Sgroi showed that v(I^k) is eventually linear with slope a(I) for k >> 0. Building on a general lower bound v(I^k) ≥ a(I)k − c(I) (Lemma 3.1), the authors propose a precise formula for all k when I has linear powers.

The conjecture extends Ficarra’s conjecture (which was posed for monomial ideals with linear powers and the formula v(I^k) = a(I)k − 1) to the broader class of graded ideals by replacing the constant 1 with c(I) = max{a(p) : p ∈ Ass(I)}. The authors verify the conjecture for several classes, including principal ideals, graded ideals with depth(S/I) = 0 whose associated primes are generated by linear forms, cover ideals with linear resolution, t-path ideals with linear powers, monomial ideals generated in degree 2 with linear resolution, edge ideals of weighted oriented graphs with linear resolution, and vertex splittable ideals, and they provide reductions for other classes (e.g., square-free monomial ideals).