Algebraic jet spaces and Zilber’s dichotomy in DCFA

Abstract

This is the first of two papers devoted to the proof of Zilber’s dichotomy for the case of difference-differential fields of characteristic zero. In this paper we use the techniques exposed in [9] to prove a weaker version of the dichotomy, more precisely, we prove the following: in DCFA the canonical base of a finite-dimensional type is internal to the fixed field of the field of constants. This will imply a weak version of Zilber’s dichotomy: a finite-dimensional type of SU-rank 1 is either 1-based or non-orthogonal to the fixed field of the field of constants.

El presente es el primero de dos art´?culos dedicados a la demostraci´on de la dicotom´?a de Zilber para el caso de los camposdifernciales de diferencia de caracter´?stica cero. En ´este art´?culo utilizamoslas t´ecnicas desarrolladas en [9] para demostrar una versi´ond´ebil de la dicotom´?a: un tipo de dimensi´on finita y de rango SUigual a 1 es modular o no ortogonal al campo fijo del campo deconstantes.