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

Authors:
Satya Deo and Lipika Mazumdar

Journal:
Math. Comp. **67** (1998), 1107-1120

MSC (1991):
Primary 41A15; Secondary 13C10

MathSciNet review:
1459386

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polyhedral complex embedded in the euclidean space and , , denote the set of all -splines on . Then is an -module where 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 -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.**M. F. Atiyah and I. G. Macdonald,*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802****2.**Louis J. Billera,*Homology of smooth splines: generic triangulations and a conjecture of Strang*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 325–340. MR**965757**, 10.1090/S0002-9947-1988-0965757-9**3.**Louis J. Billera,*The algebra of continuous piecewise polynomials*, Adv. Math.**76**(1989), no. 2, 170–183. MR**1013666**, 10.1016/0001-8708(89)90047-9**4.**Louis J. Billera and Lauren L. Rose,*Modules of piecewise polynomials and their freeness*, Math. Z.**209**(1992), no. 4, 485–497. MR**1156431**, 10.1007/BF02570848**5.**Louis J. Billera and Lauren L. Rose,*A dimension series for multivariate splines*, Discrete Comput. Geom.**6**(1991), no. 2, 107–128. MR**1083627**, 10.1007/BF02574678**6.**L. J. Billera and L. L. Rose,*Gröbner basis methods for multivariate splines*, Mathematical methods in computer aided geometric design (Oslo, 1988), Academic Press, Boston, MA, 1989, pp. 93–104. MR**1022702****7.**Satya Deo,*On projective dimension of spline modules*, J. Approx. Theory**84**(1996), no. 1, 12–30. MR**1368724**, 10.1006/jath.1996.0002**8.**Ruth Haas,*Module and vector space bases for spline spaces*, J. Approx. Theory**65**(1991), no. 1, 73–89. MR**1098832**, 10.1016/0021-9045(91)90113-O**9.**Sergey Yuzvinsky,*Cohen-Macaulay rings of sections*, Adv. in Math.**63**(1987), no. 2, 172–195. MR**872352**, 10.1016/0001-8708(87)90052-1**10.**Sergey Yuzvinsky,*Modules of splines on polyhedral complexes*, Math. Z.**210**(1992), no. 2, 245–254. MR**1166523**, 10.1007/BF02571795

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
41A15,
13C10

Retrieve articles in all journals with MSC (1991): 41A15, 13C10

Additional Information

**Satya Deo**

Affiliation:
Department of Mathematics and Computer Science, R.D. University, Jabalpur - 482 001, India

Email:
sdt@rdunijb.ren.nic.in

**Lipika Mazumdar**

Affiliation:
Department of Mathematics and Computer Science, R.D. University, Jabalpur - 482 001, India

DOI:
http://dx.doi.org/10.1090/S0025-5718-98-00943-0

Received by editor(s):
December 15, 1994

Received by editor(s) in revised form:
March 3, 1997

Additional Notes:
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

Article copyright:
© Copyright 1998
American Mathematical Society