Efficient polynomial reduction

H–bases are 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 without term orders, an important task is to determine the orthogonal projection of a homogeneous polynomial to certain subspaces of homogeneous polynomials with respect to a given inner product. One way of doing so is to use an orthogonal basis of the subspace. In this paper, we present and study a method to efficiently compute such a basis for a particular but important inner product. Dedicated to Mariano Gasca, a very special friend and collaborator, on the occasion of his sixtieth birthday