[Paper Review] A Necessary and Sufficient Condition for Local Maxima of Polynomial Modulus Over Unit Disc

Bahman Kalantari|arXiv (Cornell University)|May 2, 2016

Coding theory and cryptography4 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for local maxima of the modulus of a complex polynomial over the unit disc: a point $ z_* $ is a local maximum if and only if it equals the normalization of the Newton direction $ p(z_*)/p'(z_*) $. This characterization enables novel iterative algorithms—particularly a Pseudo-Newton method—for computing the infinity norm $ \|p\|_\infty $, with visualizations demonstrating convergence behavior via polynomiography.

ABSTRACT

An important quantity associated with a complex polynomial $p(z)$ is $\Vert p \Vert_\infty$, the maximum of its modulus over the unit disc $D$. We prove, $z_* \in D$ is a local maximum of $|p(z)|$ if and only if $a_*$ satisfies, $z_*=p(z_*)|p'(z_*)|/p'(z_*)|p(z_*)|$, i.e. it is proportional to its corresponding Newton direction. This explicit formula gives rise to novel iterative algorithms for computing $\Vert p \Vert_\infty$. We describe two such algorithms, including a Newton-like method and present some visualization of their performance.

Motivation & Objective

• To derive a precise, explicit condition for when a point on the unit disc is a local maximum of the modulus of a complex polynomial.
• To develop iterative algorithms for computing the infinity norm $ \|p\|_\infty $, which is the maximum of $ |p(z)| $ over the unit disc.
• To connect geometric properties of polynomial modulus with optimization via the Geometric Modulus Principle and Newton directions.
• To visualize convergence behavior of the proposed algorithms using polynomiography, suggesting practical and theoretical utility.

Proposed method

• Derives a necessary and sufficient condition: $ z_* = \frac{p(z_*)}{p'(z_*)} \Big/ \left| \frac{p(z_*)}{p'(z_*)} \right| $, i.e., $ z_* $ is the unit vector in the direction of the Newton step.
• Uses the Geometric Modulus Principle to analyze ascent and descent directions at a point, showing that only when the derivative index $ k=1 $ can a point be a local maximum.
• Proposes a Pseudo-Newton method to solve $ G(z) = p(z)|p'(z)| - z p'(z)|p(z)| = 0 $, where roots correspond to fixed points of the mapping $ F(z) = \frac{p(z)}{p'(z)} \Big/ \left| \frac{p(z)}{p'(z)} \right| $.
• Applies fixed-point iteration and Newton-like updates to $ G_k(z) $, using $ z_k $ as a seed to solve a linearized version at each step.
• Visualizes convergence via polynomiography, showing basins of attraction for $ F(z) $ and the Pseudo-Newton method on specific polynomials like $ z^n - 1 $.
• Suggests generalization to higher-order methods from the Basic Family for faster convergence in future work.

Experimental results

Research questions

• RQ1 What is the exact condition under which a point on the unit disc is a local maximum of $ |p(z)| $ for a complex polynomial $ p(z) $?
• RQ2 How can the Newton direction $ p(z)/p'(z) $ be used to construct iterative algorithms for computing $ \|p\|_\infty $?
• RQ3 What is the dynamical behavior of the fixed-point iteration $ F(z) = \frac{p(z)}{p'(z)} / \left| \frac{p(z)}{p'(z)} \right| $, and how does it relate to convergence to $ \|p\|_\infty $?
• RQ4 Can polynomiography effectively visualize the convergence of iterative methods for computing $ \|p\|_\infty $, and what insights does it reveal?
• RQ5 How does the Pseudo-Newton method compare in performance and convergence to standard root-finding or optimization techniques for this problem?

Key findings

• A point $ z_* \in D $ is a local maximum of $ |p(z)| $ if and only if $ z_* = \frac{p(z_*)}{p'(z_*)} \Big/ \left| \frac{p(z_*)}{p'(z_*)} \right| $, providing a precise characterization of such points.
• The condition implies that local maxima must lie on the unit circle and that $ p'(z_*) \neq 0 $, with the derivative index $ k=1 $ at such points.
• The Pseudo-Newton method for solving $ G(z) = 0 $, where $ G(z) = p(z)|p'(z)| - z p'(z)|p(z)| $, converges to fixed points corresponding to local maxima of $ |p(z)| $.
• Polynomiography of the Pseudo-Newton method reveals basins of attraction for fixed points, with small red neighborhoods near roots of $ p(z) $ acting as starting regions for convergence.
• The method shows promising dynamical behavior, with repulsive fixed points at roots of $ p(z) $ and $ p'(z) $, and attracting regions near local maxima.
• Theoretical analysis suggests that computing $ \|p\|_\infty $ is at least as hard as solving $ p(z) - r z p'(z) = 0 $ for $ r \geq 1/n $, with $ n $ the degree of $ p(z) $.