An algorithm for constructing a basis for Cr-spline modules over polynomial rings

Let 2 be a polyhedral complex embedded in the euclidean space Ed and Sr(2), r ≥ 0, denote the set of all Cr-splines on 2. Then Sr(2) is an R-module where R = E[x1, . . . , xd] is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above R-module in terms of obvious linear forms defining common faces of members of 2. This is done for the case when 2 consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms.