Ideals of powers of linear forms

Shan, Jianyun

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https://hdl.handle.net/2142/72784 Copy

Description

Title

Ideals of powers of linear forms

Author(s)

Shan, Jianyun

Issue Date

2015-01-21

Director of Research (if dissertation) or Advisor (if thesis)

Schenck, Hal

Doctoral Committee Chair(s)

D'Angelo, John

Committee Member(s)

• Schenck, Hal
• Nevins, Thomas A.
• Yong, Alexander

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2014-12

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

Copyright and License Information

Copyright 2014 Jianyun Shan

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