Discrete Mathematics & Theoretical Computer Science |

- 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 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.

Source: HAL:hal-01184777v1

Volume: DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)

Section: Proceedings

Published on: January 1, 2007

Imported on: May 10, 2017

Keywords: analytic functions,Airy phenomena,asymptotics,coalescing saddle-point method,multivariable generating functions,polynomial canonical representations,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]

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