Aequationes mathematicae

, Volume 85, Issue 3, pp 273–328

# The projective translation equation and rational plane flows. I

• Giedrius Alkauskas
Article

## Abstract

Let x = (x, y). A plane flow is a function $${F(\mathbf{x}, t) : \mathbb{R}^2 \times \mathbb{R} \mapsto \mathbb{R}^{2}}$$ 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, t). In this paper we investigate rational plane flows which are of the form $${F(\mathbf{x}, t) = \phi(\mathbf{x}t)/t;}$$ here $${\phi}$$ 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 $${\mathbb{R}}$$ all rational solutions of the two dimensional translation equation $${(1 - z)\phi(\mathbf{x}) = \phi(\phi(\mathbf{x}z)(1 - z)/z)}$$. 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 related to rational flows will be treated in the second part of this work.

## Mathematics Subject Classification (2000)

Primary 39B12 14E07 Secondary 35F05 37E35

## Keywords

Translation equation flow projective geometry rational functions rational vector fields iterable functions birational transformations involutions Cremona group linear ODE linear PDE

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