Almost Newton, sometimes Latt\`es

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.