Moment vanishing problem and positivity: Some examples

We consider the problem of vanishing of the moments m(k) (P, q) = integral(Omega) P-k(x)q(x)d mu(x) = 0, k = 1, 2, ... , with Omega a compact domain in R-n and P(x), q(x) complex polynomials in x is an element of Omega (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34,35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture: Conjecture A. For positive mu if m(k)(P, 1) = 0 for k = 1, 2, ... , then m(k)(P, q) = 0 for k >> 1 for any q. We recall recent results on one-dimensional (MVP) obtained in Muzychuk and Pakovich (2009) [24], Pakovich (2009,2004) [25,26], Pakovich (preprint) [28] and prove some initial results in several variables, stressing the role of the positivity assumption on the measure mu. On this base we analyze some special cases of Conjecture A and provide in these cases a complete characterization of the measures mu for which this conjecture holds. (C) 2010 Elsevier Masson SAS. All rights reserved.