A DIMENSION SERIES FOR MULTIVARIATE SPLINES

For a polyhedral subdivision DELTA of a region in Euclidean d-space, we consider the vector space C(k)r (DELTA) consisting of all C(r) piecewise polynomial functions over DELTA of degree at most k. We consider the formal power series SIGMA-k greater-than-or-equal-to o dim(R) C(k)r(DELTA)-lambda-k, and show, under mild conditions on DELTA, that this always has the form P(lambda)/(1 - lambda)d + 1, where P(lambda) is a polynomial in lambda with integral coefficients which satisfies P(O) = 1, P(1) = f(d)(DELTA), and P'(1) = (r + 1)f(d-1)-degrees-(DELTA). We discuss how the polynomial P(lambda) and bases for spaces C(k)r(DELTA) can be effectively calculated by use of Grobner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.