  Let P(z) and Q(y) be polynomials of the same degree k>=1 in the complex
variables z and y, respectively. In this extended abstract we study the
non-linear functional equation P(z)=Q(y(z)), where y(z) is restricted to be
analytic in a neighborhood of z=0. We provide sufficient conditions to ensure
that all the roots of Q(y) are contained within the range of y(z) as well as to
have y(z)=z as the unique analytic solution of the non-linear equation. Our
results are motivated from uniqueness considerations of polynomial canonical
representations of the phase or amplitude terms of oscillatory integrals
encountered in the asymptotic analysis of the coefficients of mixed powers and
multivariable generating functions via saddle-point methods. Uniqueness shall
prove important for developing algorithms to determine the Taylor coefficients
of the terms appearing in these representations. The uniqueness of Levinson's
polynomial canonical representations of analytic functions in several variables
follows as a corollary of our one-complex variables results.
