An E_x000E_cient Algorithm to Compute H-bases

H-bases are a particular kind of bases for polynomial ideals, characterized by the fact that their homogeneous leading terms are a basis for the associated homogeneous ideal. In the computation of H-bases, the main obstacle is to determine a system of generators for each under consideration syzygy module. The only known method to doing so, is based on the Gr¨obner bases computation which makes the computation of H-bases far from e ciency. In this paper, we present and study an e cient method to compute H-bases without relying on the Gr¨obner bases calculation.