An algorithm for constructing a basis for $C^{r}$-spline modules over polynomial rings

Let $\Box$ be a polyhedral complex embedded in the euclidean space $E^{d}$ and $S^{r}(\Box)$, $r \geq 0$, denote the set of all $C^{r}$-splines on $\Box$. Then $S^{r}(\Box)$ is an $R$-module where $R = E[x_{1},\ldots,x_{d}]$ 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 $\Box$. This is done for the case when $\Box$ consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms.