A Sequence of Nearest Polynomials with Given Factors

Abstract

Let $K$ be either $\mathbb {R}$ or $\mathbb {C}$, and $p$ and $f_0$ be polynomials in $K[x_1,\ldots ,x_s]$ such that $p\ne 0$, $\Vert f_0\Vert =1$, where $\Vert f_0\Vert$ is the Euclidean norm of $f_0$, and the coefficient of $f_0$ with the maximal absolute value is a positive real number. For $j=1$, $2$, ..., let $p_{2j-1}=f_{j-1}g_j$ be the nearest polynomial to $p$ such that $f_{j-1}|p_{2j-1}$ and $\deg (p_{2j-1})\le \deg (p)$, where $\deg$ is the total degree, and $p_{2j}=c_j f_j g_j$ be the nearest polynomial to $p$ such that $c_j\in K$, $g_j|p_{2j}$, $\deg (p_{2j})\le \deg (p)$, $\Vert f_j\Vert = 1$, and the coefficient of $f_j$ with the maximal absolute value is a positive real number. We investigate the behavior of the sequences $\{\,p_j\,\}$, $\{\,f_j\,\}$, $\{\,g_j\,\}$, and $\{\,c_j\,\}$.