Almost Newton, sometimes Lattès

Self-maps everywhere defined on the projective space over a global field are the basic objects of study in the arithmetic of dynamical systems. In this paper we study the natural self-maps defined the following way: is a homogeneous polynomial of degree in variables defining a smooth hypersurface. Suppose the characteristic of the field does not divide and define the map of partial derivatives . One can also compose such a map with an element of . In the case , the smoothness condition means that has only simple zeroes and we prove that a self-map of has constant multipliers if and only if it has the form . We recover in this manner classical dynamical systems like the Newton method for finding roots of polynomials or the Lattès map corresponding to the multiplication by 2 on an elliptic curve and the multiplication by on the multiplicative group.

在整体域上的射影空间处处有定义的自映射是动力系统算术研究的基本对象。在本文中，我们以下列方式研究自然自映射：设\(F\)是\(n + 1\)个变量\(X_0,\cdots,X_n\)的\(d\)次齐次多项式，它定义了一个光滑超曲面。假设域的特征不整除\(d\)，并定义偏导数映射\(\varphi_F = (\frac{\partial F}{\partial X_0},\cdots,\frac{\partial F}{\partial X_n})\)。也可以将这样的映射与\(PGL_{n + 1}\)的一个元素复合。在\(n = 1\)的情形下，光滑性条件意味着\(F\)只有单零点，并且我们证明\(\mathbb{P}^1\)的一个自映射具有常乘数当且仅当它具有\(\varphi_F\circ g\)的形式，其中\(g\in PGL_2\)。我们以这种方式恢复了经典动力系统，比如用于求多项式根的牛顿法，或者与椭圆曲线上乘以\(2\)以及乘法群上乘以\(2\)相对应的拉特映射。