Real Even Symmetric Forms

Harris, William Richard

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https://hdl.handle.net/2142/72534 Copy

Description

Title

Real Even Symmetric Forms

Author(s)

Harris, William Richard

Issue Date

1992

Doctoral Committee Chair(s)

Reznick, B.,

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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. 559-580.
• We present an easily-checked, necessary and sufficient condition for an even symmetric n-ary 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.

Graduation Semester

1992

Type of Resource

text

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http://hdl.handle.net/2142/72534

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