NOTES ON SOME POSITIVE SEMIDEFINITE FORMS

Abstract

Pn,m을 n원 m차 동형 다항식이라 하고(실수공간 ) ∑n,m을 2차식의 합으로 표시될 수 있는 n원 m차 동형다항식이라 하자. 3원6차 동형다항식 M, R과 4원4차 동형다항식을 다음과 같이 정의하였을때: M(x,y,z) = z^(6)+x^(4)y^(2)+x^(2)y^(4)-3x^(2)y^(2)z^(2) R (x,y,z) = x^(2)(x^(2)-z^(2))+y^2(y^(2)-z^(2))^2-(x^2-z^2)(y^2-z^2)(x^2+y^2-z^2) Q (x,y,z,w)= x^2(x-w)^2+y^2(y-w)^2+z^2(z-w)^2+2xyz( x+r+z-2w) 우리는 다음을 보였다. M ∈ P3,6 -∑3,6 R ∈ P3,6 -∑3,6 Q ∈ P4,4 -∑4,4 또한 Zero sets의 구조와, 적당한 항들의 제거로 다항식 M, R의 특성을 알아보고 M, R이 extremal 다항식임을 보였다.;Let P_(n), _(m) be the homogeneous polynomial of real forms F in n variables of degree m (^(″)n-ary m-ics^(″)) which are positive semidefinite, and let ∑_(n), _(m) be the homogeneous polynomial of n-ary m-ics which can be written as sums of squares of polynomials. For two ternary sextics M, R and a quaternary quartic Q defined as following ; M(x, y, z) = Z^(6)+x^(4)y^(2)+x^(2)^y(4)-3x^(2)y^(2)z^(2) R(x, y, z) = x^(2)(x^(2)-z^(2))^(2)+y^(2)(y^(2)-z^(2))^(2)-(x^(2)-z^(2))(y^(2)-z^(2))(x^(2)+y^(2)-z^(2)) Q(x, y, z, w) = x^(2)(x-w)^(2)+y^(2)(y-w)^(2)+z^(2)(z-w)^(2)+2xyz(x+y+z-2w), we show that M ∈ P_(3, 6) - ∑_(3, 6) R ∈ P_(3, 6) - ∑_(3, 6) Q ∈ P_(4, 4) - ∑_(4, 4) Also, it turned out that the forms M and R can be characterized by the structure of their zero sets, and by the absence of certain specific monomial terms, and we show that these are extremal forms.