Uniqueness of polynomial canonical representations

Manuel Lladser

doi:10.46298/dmtcs.3529

Abstract

Let $P(z)$ and $Q(y)$ be polynomials of the same degree $k \geq 1$ 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.

Highlights

For all |t | < r the polynomial equation in the variable td: P (t, td) = 0, with |td| < rd, has exactly k solutions repeated according to their multiplicity
The problem of whether U (t) itself can be represented as a polynomial with respect to a possibly auxiliary variable dates back to the investigations of Chester, Friedman and Ursell [CFU57] who studied this problem for the special case of d = 2
Polynomial canonical representations are pivotal for analyzing the asymptotic behavior of oscillatory integrals [BH86]

Summary

Manuel Lladser

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. 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. The uniqueness of Levinson’s polynomial canonical representations of analytic functions in several variables follows as a corollary of our one-complex variables results

Introduction

Notice that k