The Composition of Polynomials is a Determinant

Abstract

We show that the composition of two univariate polynomials can be written as a characteristic polynomial using a block-companion matrix. This matrix is useful to determine whether a polynomial is the composition of two other polynomials. Moreover, it provides bounds for the moduli of roots of such polynomials in the complex case.

MSC::

• Primary 15A15
• Secondary 15A18

ACKNOWLEDGMENT

I would like to thank Oscar F. Márquez from Universidade Federal de Santa Maria (Brazil) for his programming assistance to run the calculations of Example 1 and for useful discussions. I would also like to thank the referees for their constructive comments and suggestions that helped to improve and extend significantly the several versions of this article.

Notes

1 Since $p_{\mathit{c}}=(p_{\mathit{b}}°P^{(-1)})°(P°p_{\mathit{a}})$, where $P(\lambda )=u\lambda +v,u\ne 0$, and $P^{(-1)}(\lambda )=u^{-1}(\lambda -v)$, the condition $a_0=0$ can be achieved using $P(\lambda )=\lambda -a_0$ if necessary.

Additional information

Funding

Ministerio de Economía, Industria y Competitividad, Gobierno de España, MTM2016-77642-C2-1-P, Universidad Sergio Arboleda, IN.BG.086.20.002

Notes on contributors

Sergio A. Carrillo

Sergio A. Carrillo received his Ph.D. in mathematics in 2016 from Universidad de Valladolid (Spain) as part of the ECSING research group. After a one-year postdoc at the University of Vienna, he moved back to Bogotá where he is an assistant professor in the Mathematics Department of Universidad Sergio Arboleda. His research interests focus on summability applied to analytic problems. In his spare time, he likes to play the Spanish guitar, enjoying the mountainous landscapes that surround his hometown Zipaquirá.