The Division Algorithm in Sextic Truncated Moment Problems

다국어 초록 (Multilingual Abstract)

<P>For a degree 2n finite sequence of real numbers beta beta((2n)) = {beta(00,) beta 10, beta 01,..., beta 2n, 0, beta 2n-1,1,.., beta 1,2n-1, beta 0,2n}to have a representing measure mu, it is necessary for the associated moment matrix M(n) to be positive semidefinite, and for the algebraic variety associated to beta, nu beta nu(M(n)),, to satisfy rank M(n) <= card nu(beta) as well as the following consistency condition: if a polynomial p(x,y) equivalent to Sigma(ij) a(ij) x(4) y(i) of degree at most 2n vanishes on V beta, then the Riesz functional Lambda(p) equivalent to p(beta) := Sigma(ij) a(ij) beta(ij)=0. Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic (n=1) and quartic (n=2) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the Division Algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.</P>