# The projective translation equation and rational plane flows. I

## 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.

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## Author information

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Correspondence to Giedrius Alkauskas.

The author gratefully acknowledges support by the Lithuanian Science Council whose postdoctoral fellowship is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”.

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Alkauskas, G. The projective translation equation and rational plane flows. I. Aequat. Math. 85, 273–328 (2013). https://doi.org/10.1007/s00010-013-0190-1

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• DOI: https://doi.org/10.1007/s00010-013-0190-1

### 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