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Title:  Real Even Symmetric Forms 
Author(s):  Harris, William Richard 
Doctoral Committee Chair(s):  Reznick, B., 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Let F$\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\in$ F$\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\sbsp{n,m}{e}$ and $\Sigma$S$\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\sbsp{n,m}{e},$ respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam and B. Reznick, Math. Z. 195 (1987), pp. 559580. We present an easilychecked, necessary and sufficient condition for an even symmetric nary octic to be in PS$\sbsp{n,8}{e}$ and for an even symmetric ternary decic to be in PS$\sbsp{3,10}{e},$ and also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10. We proceed to discuss the extremal elements of the cones PS$\sbsp{3,8}{e},$ PS$\sbsp{3,10}{e}$ and PS$\sbsp{4,8}{e}.$ This leads to the question: how many of these extremal forms have sos representations? We prove that PS$\sbsp{3,8}{e}$ = $\Sigma$S$\sbsp{3,8}{e},$ a companion result to Hilbert's theorem noted above, with regard to psd ternary quartics. We also demonstrate that neither PS$\sbsp{3,10}{e}\\\Sigma$S$\sbsp{3,10}{e}$ nor PS$\sbsp{4,8}{e}\\\Sigma$S$\sbsp{4,8}{e}$ is empty, providing many new examples of psd forms which are not sos. We give a graphic representation with examples of ternary forms which also indicates whether or not an element of S$\sbsp{3,8}{e}$ or S$\sbsp{3,10}{e}$ is psd. We interpret elements of PS$\sbsp{n,m}{e}$ as inequalities; in particular, we give all symmetric polynomial inequalities of degree $\le$5 satisfied by the sides of a triangle. 
Issue Date:  1992 
Type:  Text 
Description:  104 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1992. 
URI:  http://hdl.handle.net/2142/72534 
Other Identifier(s):  (UMI)AAI9305548 
Date Available in IDEALS:  20141217 
Date Deposited:  1992 
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Dissertations and Theses  Mathematics

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