A dimension series for multivariate splines |
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Authors: | Louis J Billera Lauren L Rose |
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Institution: | (1) Cornell University, 14853-7901 Ithaca, NY, USA;(2) Ohio State University, 43210 Columbus, OH, USA |
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Abstract: | For a polyhedral subdivision Δ of a region in Euclideand-space, we consider the vector spaceC k r (Δ) consisting of allC r piecewise polynomial functions over Δ of degree at mostk. We consider the formal power series ∑ k≥0 dim? C k r (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)=f d (Δ), andP′(1)=(r+1)f d?1 0 (Δ). We discuss how the polynomialP(λ) and bases for the spacesC k r (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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