On the paths of steepest descent for the norm of a one variable complex polynomial

Damien Roy

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arXiv · 2010.01583

On the paths of steepest descent for the norm of a one variable complex polynomial

Damien Roy

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Abstract

We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial $f$. We show that such paths, starting from a zero of the logarithmic derivative of $f$ and ending in a root of $f$, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of $f$, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk.

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Damien Roy. 2020-10-04. On the paths of steepest descent for the norm of a one variable complex polynomial. https://arxiv.org/abs/2010.01583

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