Ideals of powers of linear forms

This thesis addresses two closely related problems about ideals of powers of linear forms. In the first chapter, we analyze a problem from spline theory, namely to compute the dimension of the vector space of tri-variate splines on a special class of tetrahedral complexes, using ideals of powers of linear forms. By Macaulay's inverse system, this class of ideals is closely related to ideals of fat points. In the second chapter, we approach a conjecture of Postnikov and Shapiro concerning the minimal free resolutions of a class of ideals of powers of linear forms in n variables which are constructed from complete graphs on n + 1 vertices. This statement was also conjectured by Schenck in the special case of n = 3. We provide two different approaches to his conjecture. We prove the conjecture of Postnikov and Shapiro under the additional condition that certain modules are free.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Shan, Jianyun

Contributors dc:contributor

• Schenck, Hal
• D'Angelo, John
• Nevins, Thomas A.
• Yong, Alexander

• Splines
• fat points
• free resolutions
• powers of linear forms

Statement dc:rights

• Copyright 2014 Jianyun Shan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/72784

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72784