A dimension series for multivariate splines

For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceCkr(Δ) consisting of allCr piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑k≥0 dimℝ Ckr(Δ)λk and show, under mild conditions on Δ, that this always has the formP(λ)/(1−λ)d+1, whereP(λ) is a polynomial in λ with integral coefficients which satisfiesP(0)=1,P(1)=fd (Δ), andP′(1)=(r+1)fd−10(Δ). We discuss how the polynomialP(λ) and bases for the spacesCkr(Δ) can be effectively calculated by use of Gröbner basis techniques of computational commutative algebra. A further application is given to the theory of hyperplane arrangements.

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