# A few results concerning the Schur stability of the Hadamard powers and the Hadamard products of complex polynomials

## TL;DR

This paper investigates the Schur stability of Hadamard powers and products of complex polynomials, establishing conditions and thresholds for stability and providing numerical examples to illustrate these theoretical results.

## Contribution

It introduces new criteria and thresholds for Schur stability of Hadamard powers and products of complex polynomials, expanding understanding in this area.

## Key findings

- Existence of two critical p-values determining stability regions.
- Simple sufficient conditions for Schur stability of Hadamard products.
- Numerical examples illustrating theoretical results.

## Abstract

For a complex polynomial \[ f\left( s\right) =s^{n}+a_{n-1}s^{n-1}+\ldots+a_{1}s+a_{0}% \] and for a rational number $p$, we consider the Schur stability problem of the $p$-th Hadamard power of $f$ \[ f^{\left[ p\right] }\left( s\right) =s^{n}+a_{n-1}^{p}s^{n-1}+\ldots +a_{1}^{p}s+a_{0}^{p}\text{.}% \] We show that there exist two numbers $p^{\ast}\geq0\geq p_{\ast}$ such that $f^{\left[ p\right] }$ is Schur stable for every $p>p^{\ast}$ and is not Schur stable for $p<p_{\ast}$ (or vice versa, depending on $f$). Also, we give simple sufficient conditions for the Schur stability of the Hadamard product of two complex polynomials. Numerical examples complete and illustrate the results.