Algebraic and abelian solutions to the projective translation equation

TL;DR

This paper classifies algebraic and abelian solutions to the projective translation equation, extending previous rational solutions to include flows with algebraic or abelian orbit structures, and provides explicit examples.

Contribution

It introduces a reduction algorithm for 2-homogenic rational functions and classifies all algebraic and abelian flows with algebraic orbits, including those parametrized by abelian functions.

Findings

Classified all projective flows with algebraic orbits as abelian flows.

Identified flows parametrized by abelian functions and those transformable into algebraic curves.

Provided numerous examples of algebraic, abelian, and non-abelian flows.

Abstract

Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) $(1-z)\phi(\mathbf{x})=\phi(\phi(\mathbf{x}z)(1-z)/z)$, $\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are parametrized by abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR) the orbits of these flows can be transformed into algebraic curves $x^{A}(x-y)^{B}y^{C}\equiv\mathrm{const.}$ (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines $y\equiv\mathrm{const.}$ (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.