I will present a joint work with Alicia Dickenstein and Laura Matusevich. Algebraic functions (defined as solutions to algebraic equations with general symbolic coefficients) are classically known to satisfy certain systems of linear partial differential equations with polynomial coefficients. In the talk I will consider a more general class of systems of differential equations. We prove that such systems are holonomic and that their complex holomorphic solutions have moderate growth. We also provide an explicit formula for the holonomic rank of these systems as well as bases in their spaces of complex holomorphic and Puiseux polynomial solutions.