## Abstract

Our starting point is a basic problem in Hermite interpolation theory-namely, determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem in many cases if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo-Mumford regularity of the corresponding ideals. As applications, we establish new instances of conjectures by Chudnovsky and by Demailly on the Waldschmidt constant. Moreover, we show that conjectures on the failure of the weak Lefschetz property by Harbourne, Schenck, and Seceleanu as well as by Migliore, Miró-Roig, and the first author are true asymptotically. The latter also relies on a new result for Eulerian numbers.

Original language | English |
---|---|

Pages (from-to) | 8849-8870 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 12 |

DOIs | |

State | Published - Dec 15 2019 |

### Bibliographical note

Funding Information:Received by the editors July 13, 2018, and, in revised form, April 29, 2019. 2010 Mathematics Subject Classification. Primary 13D40, 14C20, 13F20; Secondary 13D02, 14N20, 05A10. The first author was partially supported by Simons Foundation grant #317096.

Publisher Copyright:

© 2019 American Mathematical Society

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics