Moment vanishing problem and positivity: Some examples

✍ Scribed by J.P. Francoise; F. Pakovich; Y. Yomdin; W. Zhao

• Publisher: Elsevier Science
• Year: 2011
• Tongue: French
• Weight: 202 KB
• Volume: 135
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/j.bulsci.2010.06.002

✦ Synopsis

We consider the problem of vanishing of the moments m k (P , q) = Ω P k (x)q(x) dμ(x) = 0, k = 1, 2, . . . , with Ω a compact domain in R n and P (x), q(x) complex polynomials in x ∈ Ω (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) , Zhao (2010) and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:

Conjecture A. For positive μ 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 [24], Pakovich (2009,2004) [25,26], Pakovich (preprint) and prove some initial results in several variables, stressing the role of the positivity assumption on the measure μ. On this base we analyze some special cases of Conjecture A and provide in these cases a complete characterization of the measures μ for which this conjecture holds.