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Title:  Maximum Betti Numbers for a Given Hilbert Function 
Author(s):  Hulett, Heather Ann 
Doctoral Committee Chair(s):  Evans, E.G., 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\lbrack x\sb1,\..., x\sb{n}\rbrack$ where k is a field of characteristic 0. He constructed the lexsegment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lexsegment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lexsegment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above. 
Issue Date:  1993 
Type:  Text 
Description:  60 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1993. 
URI:  http://hdl.handle.net/2142/72543 
Other Identifier(s):  (UMI)AAI9411657 
Date Available in IDEALS:  20141217 
Date Deposited:  1993 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois