Ideals of powers of linear forms

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

Shan, Jianyun

关键词: Splines;    fat points;    free resolutions;    powers of linear forms;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/72784/Jianyun_Shan.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis addresses two closely related problems about ideals of powers of linear forms.In thefirst chapter, we analyze a problem from spline theory, namely to compute the dimension of thevector 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 freeresolutions 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. Weprovide twodifferent approaches to his conjecture. We prove the conjecture of Postnikov and Shapiro under the additional condition that certain modules are free.