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A Family of Ideals of Minimal Regularity and the Hilbert Series ofCr(Δ)

✍ Scribed by Hal Schenck; Mike Stillman

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
220 KB
Volume
19
Category
Article
ISSN
0196-8858

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✦ Synopsis

For a simplicial subdivision ⌬ of a region in R 2 , we analyze the dimension of the r Ž . r Ž . vector space C ⌬ of C piecewise polynomial functions splines on ⌬ of degree k at most k. We find an exact sequence which allows us to prove that the dimension w Ž . series for splines given by Billera and Rose Discrete Comput. Geom. 6 1991 , x 107᎐128 does indeed agree with the bounds on the dimension of the spline space w Ž . given by Alfeld and Schumaker Constr. Approx. 3 1987 , 189᎐197; Numer. Math.

Ž .

x 57 1990 , 651᎐661 . We give sufficient conditions for the Alfeld᎐Schumaker bounds to be attained in all degrees, where ⌬ is a two-dimensional simplicial complex. The conditions are satisfied by the class of complexes considered by Chui w Ž .

x and Wang J. Math. Anal. Appl. 94 1983 , 197᎐221 , but also by a much broader class of complexes. Furthermore, for conditions which involve only local geometric data, this result is the strongest possible.

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