Basic Tools for Computing in Multigraded Rings

In this paper we study ℤm-gradings on the polynomial ring>P = K[x₁,...,x_nover a field K which are suitable for developing algorithms which take advantage of the full amount of homogeneity contained in a given problem. After introducing and characterizing weakly positive and positive gradings, we provide the basic properties of Macaulay bases and multihomogenization with respect to such gradings as well as the connection between these notions. Finally, we formulate the multihomogeneous version of the Buchberger algorithm for computing homogeneous Grobner bases and minimal homogeneous systems of generators.