Ficarra–Sgroi conjecture on v-numbers for ideals with linear powers

Prove that for a homogeneous ideal I in a polynomial ring over a field that has linear powers, the v-number satisfies v(I^k) = α(I)·k − 1 for all integers k ≥ 1, where α(I) denotes the initial degree of I.

Background

The paper recalls general results on the asymptotic linearity of the v-number for powers of homogeneous ideals and cites work showing that v(I^k) is eventually linear. In this context, Ficarra and Sgroi formulated a precise prediction for the exact linear form when the ideal has linear powers.

They also provided affirmative evidence for this conjecture in three specific classes: edge ideals with linear resolutions, polymatroidal ideals, and Hibi ideals. The general case beyond these classes is not settled within the cited works, and the present paper focuses on generalized binomial edge ideals rather than directly resolving the conjecture.