Almost Newton, sometimes Latt\`es

TL;DR

This paper explores a class of natural self-maps on projective hypersurfaces defined via derivatives of homogeneous polynomials, connecting classical dynamical systems like Newton's method and Lattès maps to algebraic geometry.

Contribution

It introduces and studies a family of self-maps derived from derivatives of homogeneous polynomials, linking them to classical dynamical systems and providing new insights into their fixed points and properties.

Findings

Fixed points are easily described for these maps.

The maps include classical systems like Newton's method and Lattès maps.

The smoothness of hypersurfaces relates to the non-vanishing of derivatives.

Abstract

Self-maps everywhere defined on the projective space $\P^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with complements in \cite{Bhatnagar}) that asserts that a "polarized" self-map of a projective variety is essentially the restriction of a self-map of the projective space given by the polarization. In this paper we study the natural self-maps defined the following way: $F$ is a homogeneous polynomial of degree $d$ in $(N+1)$ variables $X_i$ defining a smooth hypersurface. Suppose the characteristic of the field does not divide $d$ and define the map of partial derivatives $\phi_F = (F_{X_0},...,F_{X_N})$. The map $\phi_F$ is defined everywhere due to the following formula of Euler: $\sum X_i F_{X_i} = d F$, which implies that a point where all the partial derivatives vanish is a non-smooth point of the hypersuface F=0. One can also compose such a map with an element of $\PGL_{N+1}$. In the particular case addressed in this article, N=1, the smoothness condition means that $F$ has only simple zeroes. In this manner, fixed points and their multipliers are easy to describe and, moreover, with a few modifications we recover classical dynamical systems like the Newton method for finding roots of polynomials or the Latt\`es map corresponding to the multiplication by 2 on an elliptic curve.