Ideals of powers of linear forms
type: CreativeWork
description: 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.
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.
author: Shan, Jianyun
language: en
keywords: Splines, fat points, free resolutions, powers of linear forms
date published: 2015-01-21T19:48:10Z
date modified: 2026-07-22T22:26:07Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/72784