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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 $p$, weconsider 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 stablefor 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 Schurstability of the Hadamard product of two complex polynomials. Numericalexamples complete and illustrate the results.

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