# Algebraic and abelian solutions to the projective translation equation

## Abstract

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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## References

1. Aczél J.: Lectures on Functional Equations and Their Applications. Vol. 19. Mathematics in Science and Engineering, Academic Press, New York (1966)

2. Alkauskas G.: Multi-variable translation equation which arises from homothety. Aequ. Math. 80(3), 335–350 (2010)

3. Alkauskas G.: The projective translation equation and rational plane flows. I. Aequ. Math. 85(3), 273–328 (2013)

4. Alkauskas G.: The projective translation equation and unramified 2-dimensional flows with rational vector fields. Aequ. Math. 89(3), 873–913 (2015)

5. Alkauskas, G.: Commutative projective flows. arXiv:1507.07457

6. Beukers F.: Hypergeometric functions, how special are they?. Not. Am. Math. Soc. 61(1), 48–56 (2014)

7. Bacher R., Flajolet P.: Pseudo-factorials, elliptic functions, and continued fractions. Ramanujan J. 21(1), 71–97 (2010)

8. Flajolet, P., Gabarró, J., Pekari, H.: Analytic urns. Ann. Probab. 33(3), 1200–1233 (2005). arXiv:math/0407098

9. van Fossen Conrad, E., Flajolet, P.: The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion. Sém. Lothar. Combin. 54, (2005/07), Art. B54g. arXiv:math/0507268

10. Lang S.: Introduction to Algebraic and Abelian Functions. Graduate Texts in Mathematics, 89, 2nd edn. Springer, New York (1982)

11. Milne, J.: Algebraic geometry, v6.00. http://www.jmilne.org/math/CourseNotes/ag.html

12. Moszner Z.: The translation equation and its application. Demonstr. Math. 6, 309–327 (1973)

13. Moszner Z.: General theory of the translation equation. Aequ. Math. 50(1-2), 17–37 (1995)

14. Mumford, D.: Tata lectures on theta. I, II, III. Progress in Mathematics, 28. Birkhäuser (2007)

15. Nikolaev, I., Zhuzhoma, E.: Flows on 2-Dimensional Manifolds: An Overview. Lecture Notes in Mathematics, 1705. Springer, Berlin (1999)

16. The Online Encyclopedia of Integer Sequences, A092676, A092677. https://oeis.org/

17. Vidūnas R.: Darboux evaluations of algebraic Gauss hypergeometric functions. Kyushu J. Math. 67(2), 249–280 (2013)

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Correspondence to Giedrius Alkauskas.