On the $\mathrm{v}$-number of Gorenstein ideals and Frobenius powers

In this paper, we show the equality of the (local) $\mathrm{v}$-number and Castelnuovo-Mumford regularity of certain classes of Gorenstein algebras, including the class of Gorenstein monomial algebras. Also, for the same classes of algebras with the assumption of level, we show that the (local) $\mathrm{v}$-number serves as an upper bound for the regularity. Moreover, we investigate the $\mathrm{v}$-number of Frobenius powers of graded ideals in prime characteristic setup. In this study, we demonstrate that the $\mathrm{v}$-numbers of Frobenius powers of graded ideals have an asymptotically linear behaviour. In the case of unmixed monomial ideals, we provide a method for computing the $\mathrm{v}$-number without prior knowledge of the associated primes.