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Title:Ideals of powers of linear forms
Author(s):Shan, Jianyun
Director of Research:Schenck, Hal
Doctoral Committee Chair(s):D'Angelo, John
Doctoral Committee Member(s):Schenck, Hal; Nevins, Thomas A.; Yong, Alexander
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:2015-01-21
Rights Information:Copyright 2014 Jianyun Shan
Date Available in IDEALS:2015-01-21
Date Deposited:2014-12

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