2026-09-22T03:52:25Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/72784 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Schenck, Hal D'Angelo, John Schenck, Hal Nevins, Thomas A. Yong, Alexander Shan, Jianyun 2015-01-21T19:48:10Z 2015-01-21T19:48:10Z 2014-12 2015-01-21 2014-12 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. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-12-03T22:54:45Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 5 jimmy-bib.bib: 21548 bytes, checksum: 004a74d57b30585bf49a6a07535aa26e (MD5) Postnikov-Shapiro-conjecture.tex: 147447 bytes, checksum: e59bf00c9a333deab936235758158bcc (MD5) spline-revision.tex: 73221 bytes, checksum: c904bbb8d08b7c44bfd490d9052b899b (MD5) thesis-ex.tex: 16463 bytes, checksum: 0faad5fd0ab777e8cb7051a4e4c0aef0 (MD5) Shan_Jianyun.pdf: 648775 bytes, checksum: 8edc226928fdda536f7e9d25ae893a3b (MD5) Made available in DSpace on 2015-01-21T19:48:10Z (GMT). No. of bitstreams: 5 Jianyun_Shan.pdf: 648775 bytes, checksum: 8edc226928fdda536f7e9d25ae893a3b (MD5) jimmy-bib.bib: 21548 bytes, checksum: 004a74d57b30585bf49a6a07535aa26e (MD5) Postnikov-Shapiro-conjecture.tex: 147447 bytes, checksum: e59bf00c9a333deab936235758158bcc (MD5) spline-revision.tex: 73221 bytes, checksum: c904bbb8d08b7c44bfd490d9052b899b (MD5) thesis-ex.tex: 16463 bytes, checksum: 0faad5fd0ab777e8cb7051a4e4c0aef0 (MD5) http://hdl.handle.net/2142/72784 en Copyright 2014 Jianyun Shan Splines fat points free resolutions powers of linear forms Ideals of powers of linear forms text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD