Generalising the divisibility relation of terms we introduce the lattice of so-called involutive divisions and define the admissibility of such an involutive division for a given set of terms. Based on this theory we present a new approach for building 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 flexible 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 Gröbner basis and that it is straightforward to compute the (affine) Hilbert function of an ideal I from an arbitrary involutive basis of I.

We consider the check of the involutive basis property in a polynomial context. In order to show that a finite generating set F of a polynomial ideal I is an involutive basis one must confirm two properties. Firstly, the set of leading terms of the elements of F has to be complete. Secondly, one has to prove that F is a Gröbner basis of I. The latter is the time critical part but can be accelerated by application of Buchberger's criteria including the many improvements found during the last two decades. Gerdt and Blinkov (Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, pp. 519-541, 1998) were the first who applied these criteria in involutive basis computations. We present criteria which are also transfered from the theory of Gröbner bases to involutive basis computations. We illustrate that our results exploit the Gröbner basis theory slightely more than those of Gerdt and Blinkov. Our criteria apply in all cases where those of Gerdt/Blinkov do, but we also present examples where our criteria are superior. Some of our criteria can be used also in algebras of solvable type, e. g., Weyl algebras or enveloping algebras of Lie algebras, in full analogy to the Gröbner basis case. We show that the application of criteria enforces the termination of the involutive basis algorithm independent of the prolongation selection strategy.

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