Combinatorial Aspects of Multivariate Splines

Abstract

Let X denote an s x n integer matrix with columns x ¹ ,…,xⁿ Î Z^S \ {0}. (Sometimes we will also denote the set of vectors {x¹,…,xⁿ} by X too.) We will assume throughout the following discussion that (1.1)

$$0 \notin [X],$$

((1.1))

where [ A ] will mean the convex hull of the set A. The main object of our study is the function

$$t(a|X) = |\{ \beta \in Z_ + ^n:X\beta = a\} |,$$

((1.2))

where a E Zs and I A I denotes the cardinality of the set A.This function which we have referred to earlier as the “discrete truncated power”, [DM1], counts the number of nonnegative integer solutions a = (019 ...,pn) to the linear diophantine equations

$$\sum\limits_{j = l}^n {x_i^j{\beta _j} = {\alpha _i},{\text{ }}{\alpha _i} \in Z,i = 1, \ldots s,} $$

whose coefficient matrix is X.