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

Abstract

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 C 2 \ {union of curves}) projective flows whose vector field is still rational. If an orbit of the flow is given by 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, these 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.

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

“…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.…”

“…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. Open directions in the investigations of the projective translation equation, indicated in ( [4], Sect. 5.1.)…”

“…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.…”

“…For example, there exists exactly 5 superflows over R in dimension 3: tetrahedral (symmetry group is of order 24), octahedral (24), icosahedral (60), 4-antiprismal (16), and 3-prismal (12). The theory of unramified flows was begun in [4], and is under development in [6]. vii) Finally, now the case of rational flows in dimension 3 looks much more tractable and optimistic, if we manage to find and analogue of Theorem 2 (see Subection 2.1) and prove everything by the same "bootstrapping" method; see Subsection 4 for an inductive construction of algebraic or rational flows in any dimension.…”