Manuel Lladser

Uniqueness of polynomial canonical representations
dmtcs:3529 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2007,
DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)

https://doi.org/10.46298/dmtcs.3529
Uniqueness of polynomial canonical representations
Authors: Manuel Lladser ^{1}
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Manuel Lladser
1 Department of Applied Mathematics
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 nonlinear 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 nonlinear 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 saddlepoint 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 onecomplex variables results.