  Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation
p(z) =0, in terms of its vector of complex coefficients a, are classically
known to satisfy holonomic systems of linear partial differential equations
with polynomial coefficients. In this paper we investigate one of such systems
of differential equations which was introduced by Mellin. We compute the
holonomic rank of the Mellin system as well as the dimension of the space of
its algebraic solutions. Moreover, we construct explicit bases of solutions in
terms of the roots of p and their logarithms. We show that the monodromy of the
Mellin system is always reducible and give some factorization results in the
univariate case.
