Uniqueness of polynomial canonical representations

TL;DR

This paper investigates conditions under which a polynomial functional equation has a unique analytic solution, with implications for polynomial canonical representations in asymptotic analysis of oscillatory integrals.

Contribution

It provides sufficient conditions for the uniqueness of solutions to polynomial functional equations, extending to Levinson's polynomial canonical representations in several variables.

Findings

All roots of Q(y) are within the range of y(z).

Y(z)=z is the unique analytic solution under certain conditions.

Results aid in developing algorithms for Taylor coefficient determination.

Abstract

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.