7 problems

Let $p$ be a degree-$n$ univariate complex polynomial, and let its DR-circles be the circles centered at the roots of $p'$ with the corresponding radii $\rho_j$. For a root $z_i$ o…

Let $p$ be a degree-$n$ univariate complex polynomial. Let its DR-circles be the circles centered at the roots of $p'$ with radii $\rho_j$, where … For each root of $p$, consider i…

Fast-convergence basin conjecture. There is a universal constant $\eta_1>0$ such that, for every simple root $z_i$, some DR-circle contains a segment of length $\eta_1/n$ radians l…

Let $K$ be a finite extension of the rational field $\mathbb{Q}$ or of the rational function field $\mathbb{F}_p(T)$. Let $(x_n)$ be the Newton approximation sequence associated to…

Let $K$ be a number field, let $f\in K[x]$ be a squarefree polynomial of degree at least $2$, and let $(x_n)$ be the Newton approximation sequence associated to $f$ and an initial…

Let $f$ be a polynomial of degree $d\geq 2$ with coefficients in a number field $K$, and let $x_0\in K$. Define the Newton map … and, for each $n\geq 0$, set … Assume that the Newt…

Let $f$ be a polynomial over a number field $K$, let $x_0\in K$ be a sufficiently general starting point, and let $(x_n)$ be the Newton iteration for $f$ starting at $x_0$. For a p…