A mapping defined by the Schur-Szeg\H{o} composition

TL;DR

This paper explores the properties of a specific affine mapping related to the Schur-Szeg ext{"o} composition of polynomials, extending classical results like Descartes' rule to exponential-polynomial functions.

Contribution

It introduces and analyzes a new affine mapping associated with Schur-Szeg ext{"o} composition and extends Descartes' rule to exponential-polynomial functions.

Findings

Characterization of the affine mapping _{n,k}

Extension of Descartes' rule to exponential-polynomial functions

Properties of the mapping for functions of the form e^xP

Abstract

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szeg\H{o} composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine mapping $\Phi _{n,k}$~:~$(c_1,\ldots ,c_n)$ $\mapsto$ $(\sigma _1, \ldots ,\sigma _n)$, where $\sigma _i$ are the elementary symmetric polynomials of the numbers $a_j$. We study also properties of a similar mapping for functions of the form $e^xP$, where $P$ is a polynomial, $P(0)=1$, and we extend the Descartes rule to them.