F. Pakovich
Laurent polynomial moment problem: a case study.
Fri, 10/16/2009 - 04:10 — SomNambulAuthors: F. Pakovich, C. Pech, A. Zvonkin
Subjects: Complex Variables
AbstractIn recent years, the so-called polynomial moment problem, motivated by the
classical Poincare center-focus problem, was thoroughly studied, and the
answers to the main questions have been found. The study of a similar problem
for rational functions is still at its very beginning. In this paper, we make
certain progress in this direction; namely, we construct an example of a
Laurent polynomial for which the solutions of the corresponding moment problem
behave in a significantly more complicated way than it would be possible for a
polynomial.On rational functions orthogonal to all powers of a given rational function on a curve.
Tue, 10/13/2009 - 18:46 — SomNambulAuthors: F. Pakovich
Subjects: Complex Variables
AbstractIn this paper we study the generating function f(t) for the sequence of the
moments \int_{\gamma}P^i(z)q(z)d z, i\geq 0, where P(z),q(z) are rational
functions of one complex variable and \gamma is a curve in C. We calculate an
analytical expression for f(t) and provide conditions implying the rationality
and the vanishing of f(t). In particular, for P(z) in generic position we give
an explicit criterion for a function q(z) to be orthogonal to all powers of
P(z).Generalized "second Ritt theorem" and explicit form of solutions of the polynomial moment problem.
Wed, 08/19/2009 - 19:30 — SomNambulAuthors: F. Pakovich
Subjects: Dynamical Systems
AbstractIn the recent paper arXiv:0710.4085 was shown that any solution of so called
polynomial moment problem, which asks to describe polynomials Q orthogonal to
all powers of a given polynomial P on a segment, may be obtained as a sum of
some "reducible" solutions related to "compositional right factors" of P.
However, the methods of arXiv:0710.4085 do not permit to estimate the number of
necessary reducible solutions and their explicit form.
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