Solution of the truncated moment problem with variety 𝑦=𝑥³

Abstract

Abstract. We show that positivity, consistency, and the variety condition, which are sufficient to solve the truncated moment problem on planar curves of degree 2, are not sufficient for curves of higher degree. Using new, partly algorithmic, conditions based on positive moment matrix extensions, we present a concrete solution to the truncated moment problem on the curve y = x 3 . We also use moment matrix extensions to solve (in a less concrete sense) truncated moment problems on curves of the form y = g(x) and yg(x) = 1 (g ∈ R[x]), leading to degree-bounded weighted sum-of-squares representations for polynomials which are strictly positive on such curves.

Positivity of Riesz functionals and solutions of quadratic and quartic moment problems

“…In [13] we showed that in M 4 (ỹ) we must have y 44 ψ(y), and [13,Theorem 2.4] implies that M has a representing measure if and only if y 15 ≡ s > ψ(y). A calculation shows that for M as in (2.1) and satisfying (2.2), with appropriate values of the moment data we can also have ψ(y) independent of s and t. This is the case, for example, if we modify (2.3) so that x = 1 10 , r = 600, s is arbitrary and t is chosen sufficiently large so as to preserve positivity and the property rank M 3 (y) = 9.…”

“…3 1 }, and thus positive. Since y 1,5 = ψ(y), positivity for L y cannot be derived from the existence of a representing measure, since [13,Theorem 2.4] shows that y has no representing measure. Moreover, positivity for L y cannot be established from the positivity of M via sums-of-squares arguments because, by Hilbert's theorem, there exist degree 6 bivariate polynomials that are everywhere nonnegative but are not sums of squares.…”

“…For the case when K is a closed semialgebraic set, analogues of the preceding results appear in [8]. The papers [7,9,13] describe various concrete existence theorems for representing measures based on flat extensions. These results usually assume that M d (y) is positive semidefinite and singular, so that any representing measure is necessarily supported in the nontrivial algebraic variety V(M d (y)).…”

Sextic Moment Problems on 3 Parallel Lines

“…Beyond this order, the situation gets more complicated; many instances show a solution must include numerical conditions involving moments (see [6], [7], and [9]). …”

“…We know moment problems with a single cubic column relation or with an invertible moment matrices are much more difficult to solve since these cases naturally satisfy all the necessary conditions and require some other properties; for m = 6, some specific cases are resolved as in [9,10] but most cases remain open. In this note we focus on sextic moment problems with a single cubic column relation of 3 parallel lines, that is, rank M (n) = 9.…”