The Theory of

Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and de ne the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building up a general theory of involutive bases of polynomial ideals. In particular, we give algorithms for checking the involutive basis property and for completing an arbitrary basis to an involutive one. It turns out that our theory is more constructive and more exible than the axiomatic approach to general involutive bases due to Gerdt and Blinkov. Finally, we show that an involutive basis contains more structural information about the ideal of leading terms than a Gr obner basis and that it is straight forward to compute the (a ne) Hilbert function of an ideal I from an arbitrary involutive basis of I.