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

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 $p^{\ast }\ge 0\ge p_{\ast }$ 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.