The system of Weierstrass polynomials, defined originally for ideals in convergent power series rings, together with its sequence of degrees allows us to analyze a homogeneous ideal directly. Making use of it, we study local cohomology modules, syzygies, and then graded Buchsbaum rings. Our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of Weierstrass polynomials, a rough classification of graded Buchsbaum rings in the general case and a complete classification of graded Buchsbaum integral domains of codimension two.