The projective translation equation and unramified 2-dimensional flows with rational vector fields

Abstract

Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1−z)ϕ(x) = ϕ(ϕ(xz)(1−z)/z); here ϕ(x) = (u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in \({\mathbb{C}^{2}\setminus\{\text{union of curves}\}}\)) projective flows whose vector field is still rational. If an orbit of the flow is given by a homogeneous rational function of degree N, then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, unramified non-singular flows are of 6 types: (1) the identity flow; (2) one flow for each non-negative integer N—these flows are rational of level N; (3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; (4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; (5) the level 4 flow expressable in terms of lemniscatic elliptic functions; (6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya–Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.