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The projective translation equation and rational plane flows. I


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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Correspondence to Giedrius Alkauskas.

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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).

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Mathematics Subject Classification (2000)

  • Primary 39B12
  • 14E07
  • Secondary 35F05
  • 37E35


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