The projective translation equation and rational plane flows. I

TL;DR

This paper classifies all rational solutions to a specific projective translation equation in two dimensions, revealing a finite set of canonical flows characterized by their level and symmetry properties.

Contribution

It provides a complete classification of rational plane flows satisfying the translation equation, up to conjugation, and introduces the concept of flow level as a key invariant.

Findings

Identified all rational solutions as a finite set of canonical flows.

Classified flows by their level and symmetry properties.

Established the structure of flows under conjugation with birational transformations.

Abstract

Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F might not be well-defined for certain x,y,t). In this paper we investigate rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of rational functions in 2 real variables. These may be called projective flows, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over R all rational solutions of the two dimensional translation equation (1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer N, called the level of the flow. The case N=0 stands apart, while the case N=1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2, respectively. Many more questions will be treated in the second part of this work.