## Abstract

We explore the connection between ideals of fat points (which correspond to subschemes of P^{n}obtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inR^{d}. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals ink[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 Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree.

Original language | English (US) |
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Pages (from-to) | 116-128 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 204 |

Issue number | 1 |

DOIs | |

State | Published - Jun 1 1998 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory