MATHEMATICS OF COMPUTATION Volume 67, Number 223, July 1998, Pages 1107–1120 S 0025-5718(98)00943-0 AN ALGORITHM FOR CONSTRUCTING A BASIS FOR C r -SPLINE MODULES OVER POLYNOMIAL RINGS SATYA DEO AND LIPIKA MAZUMDAR Abstract. Let ✷ be a polyhedral complex embedded in the euclidean space E d and S r (✷), r ≥ 0, denote the set of all C r -splines on ✷. Then S r (✷) is an R-module where R = E[x 1 ,...,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 ✷. This is done for the case when ✷ consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms. 1. Introduction Let ✷ be a polyhedral d-complex embedded in the euclidean space E d , i.e., a compact subset of E d subdivided into a finite collection of d-dimensional convex polyhedra which are properly joined (see [7] for details). Fix an integer r ≥ 0. Let C r (✷) denote the set of real valued functions f defined on ✷ such that f | σ is in the polynomial ring R = E[x 1 ,...,x d ] for each d-face σ of ✷ and f is r-times continuously differentiable on the whole of ✷. The elements of C r (✷), known as multivariate splines of smoothness r, have proved extremely useful in obtaining approximate solutions of partial differential equations by finite element methods. The set C r k (✷) of all those C r -splines which are of degree ≤ k, form a real vector space. These vector spaces which are easily computable are usually taken as the approximant spaces for various suitable degrees k. Evidently these are finite dimensional. Determining their dimension as well as a basis having minimal support has been a very interesting and sometimes a difficult proposition of practical importance (see [5], [6], and [7]). The difficulty is caused by the fact that the vector space dimension of C r k (✷) depends not only on the combinatorics but also on the geometry of ✷. In order to tackle the above “dimension problem”, more algebraic approaches using the method of commutative algebra have been recently initiated by Haas [8], Billera and Rose ([3], [5], [6]). With pointwise operations of addition and multiplication the set C r (✷) forms a ring and the polynomial ring R is just a subring of C r (✷). Hence C r (✷) is an R-module in a natural way. It is easily seen that this module is finitely generated, torsion free and of rank equal to the number of d-faces of ✷. The general question as to under what condition on d, r and ✷, Received by the editor December 15, 1994 and, in revised form, March 3, 1997. 1991 Mathematics Subject Classification. Primary 41A15, Secondary 13C10. The first author was supported by the UGC research project no. F 8-5/94 (SR-I). The second author was supported by the CSIR(JRF) no. 9/97(36)/92/EMR-I. c 1998 American Mathematical Society 1107 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use