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Fat Points, Inverse Systems, and Piecewise Polynomial Functions

✍ Scribed by Anthony V Geramita; Henry K Schenck

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
181 KB
Volume
204
Category
Article
ISSN
0021-8693

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✦ Synopsis

We explore the connection between ideals of fat points which correspond to n Ž . . subschemes of ‫ސ‬ obtained by intersecting mixed powers of ideals of points , and Ž . piecewise polynomial functions splines on a d-dimensional simplicial complex ⌬ d w x embedded in R . Using the inverse system approach introduced by Macaulay 11 , we give a complete characterization of the free resolutions possible for ideals in w x Ž k x, y generated by powers of homogeneous linear forms we allow the powers to . differ . We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed ⌬, the dimension of the vector space of splines on ⌬ of degree less than or equal to k. We use this relationship and the results above to derive a formula which gives the number of Ž . planar mixed splines in sufficiently high degree.

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