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Algebraic and abelian solutions to the projective translation equation


Let \({{\mathrm x}=(x,y)}\). A projective two-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) \({(1-z)\phi({\mathrm x})=\phi(\phi({\mathrm x}z)(1-z)/z)}\), \({\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}}\). Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field \({\varpi(x,y)\bullet \varrho(x,y)}\) which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, such as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows us to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are described in terms of abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR), and the orbits of these flows can be transformed into algebraic curves \({x^{A}(x-y)^{B}y^{C}\equiv{\mathrm{const.}}}\) (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines \({y\equiv{\mathrm{const.}}}\) (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.

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Correspondence to Giedrius Alkauskas.

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The research of the author was supported by the Research Council of Lithuania Grant No. MIP-072/2015.

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Alkauskas, G. Algebraic and abelian solutions to the projective translation equation. Aequat. Math. 90, 727–763 (2016).

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Mathematics Subject Classification

  • Primary 39B12
  • 14H05
  • 14K20
  • Secondary 35F05
  • 37E35


  • Translation equation
  • Projective flow
  • Rational vector fields
  • Iterable functions
  • Linear ODE
  • Linear PDE
  • Abelian functions
  • Algebraic functions