Algebraic and abelian solutions to the projective translation equation

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.Comment: 31 pages, 5 figure

The projective translation equation and rational plane flows. II. Corrections and additions

“…The existence of rational solution to this ODE for rational projective flows was the consequence of reduction algorithm, which, as we will shortly explain, contains a serious flaw. However, while working on paper [8] it emerged that (7) plays a much deeper role in describing the flow itself. For example, Theorem 2 immediately implies that for algebraic flow (and for its special case, rational flow), all solutions to this ODE are algebraic.…”

“…are valid, except for the step I. which contains a logical flaw. ii) All the special examples of projective flows in [3,4,8], which make the majority of the text, are valid. This includes rational, algebraic, unramified, abelian flows, and one non-abelian flow with a vector field x 2 + xy + y 2 • xy + y 2 ; another class of non-abelian flows was named pseudo-flows of level 0 and was described in [3], Subsection 5.2.…”

“…To prove this and all other classification Theorems in [3,4,8], many ingredients are needed. One of the steps is the algorithm for reducing rational vector fields.…”

“…Note also that the flow with the vector field x 2 − xy • y 2 − 2xy, as given by ([4], Theorem 1) is indeed unramified. This vector field is given an extensive treatment in the Appendix of [8]. iii) Four classification Theorems (Theorem in [3], Theorem 1 in [4], Theorem 2 and Theorem 3 in [8]) remain valid as stated, if we additionally assume that with a help of a 1-BIR a rational vector field can be reduced to a pair of quadratic forms, or to a vector field when one of the components vanishes.…”

“…This vector field is given an extensive treatment in the Appendix of [8]. iii) Four classification Theorems (Theorem in [3], Theorem 1 in [4], Theorem 2 and Theorem 3 in [8]) remain valid as stated, if we additionally assume that with a help of a 1-BIR a rational vector field can be reduced to a pair of quadratic forms, or to a vector field when one of the components vanishes. As it turns out now, the last happens exactly for abelian flows of level 1; see Definition 1. iv) After corrections and additions of the current paper, we can say the following.…”