On the v-number of Gorenstein Ideals and Frobenius Powers

n this paper, we show the equality of the (local) v-number and Castelnuovo-Mumfordregularity of certain classes of Gorenstein algebras, including the class of Gorensteinmonomial algebras. Also, for the same classes of algebras with the assumption oflevel, we show that the (local) v-number serves as an upper bound for the regularity.As an application, we get the equality between the v-number and regularity for Stanley-Reisner rings of matroid complexes. Furthermore, this paper investigates the v-numberof Frobenius powers of graded ideals in prime characteristic setup. In this direction,we demonstrate that the v-numbers of Frobenius powers of graded ideals have anasymptotically linear behaviour. In the case of unmixed monomial ideals, we providea method for computing the v-number without prior knowledge of the associatedprimes.