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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Interpolation and the weak Lefschetz property


Authors: Uwe Nagel and Bill Trok
Journal: Trans. Amer. Math. Soc. 372 (2019), 8849-8870
MSC (2010): Primary 13D40, 14C20, 13F20; Secondary 13D02, 14N20, 05A10
DOI: https://doi.org/10.1090/tran/7889
Published electronically: August 20, 2019
MathSciNet review: 4029714
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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.


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Additional Information

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email: uwe.nagel@uky.edu

Bill Trok
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email: william.trok@uky.edu

DOI: https://doi.org/10.1090/tran/7889
Received by editor(s): July 13, 2018
Received by editor(s) in revised form: April 29, 2019
Published electronically: August 20, 2019
Additional Notes: The first author was partially supported by Simons Foundation grant #317096.
Article copyright: © Copyright 2019 American Mathematical Society