Polynomiality of the Generalized Verschiebung Degree

For a general curve in positive characteristic, taking the Frobenius pullback induces a generically finite rational map V on the moduli space of rank 2 vector bundles with trivial determinant. Recently, Kondo–Wakabayashi show that the generic degree of V, considered as a function on the characteristic of the base field, is a quasi-polynomial. In this paper, we show that this quasi-polynomial is indeed a polynomial, and we write out this polynomial explicitly.