Aequationes mathematicae

, Volume 89, Issue 3, pp 873–913

# The projective translation equation and unramified 2-dimensional flows with rational vector fields

• Giedrius Alkauskas
Article

## Abstract

Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1−z)ϕ(x) = ϕ(ϕ(xz)(1−z)/z); here ϕ(x) = (u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in $${\mathbb{C}^{2}\setminus\{\text{union of curves}\}}$$) projective flows whose vector field is still rational. If an orbit of the flow is given by a homogeneous rational function of degree N, then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, unramified non-singular flows are of 6 types: (1) the identity flow; (2) one flow for each non-negative integer N—these flows are rational of level N; (3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; (4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; (5) the level 4 flow expressable in terms of lemniscatic elliptic functions; (6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya–Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.

## Keywords

Projective translation equation flows rational vector fields iterative functional equation elliptic curves elliptic functions Dixonian elliptic functions linear PDE’s finite group representations hypergeometric functions

## Mathematics Subject Classification

Primary 39B12 37E35 Secondary 14H52 14H05 14E05

## References

1. 1.
Aczél, J.: Lectures on functional equations and their applications. Mathematics in Science and Engineering, Vol. 19 Academic Press, New York-London (1966)Google Scholar
2. 2.
Ahiezer, N.I.: Èlementy teorii èlliptičeskih funkciĭ. (Russian) [Elements of the Theory of Elliptic Functions] Gosudarstv. Izdat. Tehn. Teor. Lit., Moscow-Leningrad,. 291 pp (1948)Google Scholar
3. 3.
Alkauskas G.: Multi-variable translation equation which arises from homothety. Aequ. Math. 80(3), 335–350 (2010)
4. 4.
Alkauskas G.: The projective translation equation and rational plane flows. I. Aequ. Math. 85(3), 273–328 (2013)
5. 5.
Alkauskas, G.: The projective translation equation and rational plane flows. II (in preparation)Google Scholar
6. 6.
Alkauskas, G.: The projective translation equation: rational vector fields and quasi-flows (tentative title)Google Scholar
7. 7.
Bacher R., Flajolet Ph.: Pseudo-factorials, elliptic functions, and continued fractions. Ramanujan J. 21(1), 71–97 (2010)
8. 8.
Blasiak, P., Flajolet, Ph.: Combinatorial models of creation-annihilation. Sém. Lothar. Combin. 65, Art. B65c (2011)Google Scholar
9. 9.
Dixon, A.C.: On the doubly periodic functions arising out of the curve x 3y 3 − 3αxy = 1. Quart. J. XXIV, 167–233 (1890)Google Scholar
10. 10.
Dumont D.: A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions. Math. Comp. 33, 1293–1297 (1979)
11. 11.
Dumont D.: Une approche combinatoire des fonctions elliptiques de Jacobi. Adv. Math. 1, 1–39 (1981)
12. 12.
Dumont, D.: Grammaires de William Chen et dérivations dans les arbres et arborescences. Sém. Lothar. Combin. 37, Art. B37a, 21 pp. (electronic) (1996)Google Scholar
13. 13.
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Francesco, G.: Higher transcendental functions. Vol. I. Based on notes left by Harry Bateman. With a preface by Mina Rees. With a foreword by Watson, E.C. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla. (1981)Google Scholar
14. 14.
Flajolet Ph., Gabarró J., Pekari H.: Analytic urns. Ann. Probab. 33(3), 1200–1233 (2005)
15. 15.
van Fossen Conrad, E., Flajolet, Ph.: The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion. Sém. Lothar. Combin. 54 , Art. B54g (2005/07)Google Scholar
16. 16.
Flajolet, Ph., Dumas, Ph., Puyhaubert, V.: Some exactly solvable models of urn process theory. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, Discrete Math. Theor. Comput. Sci. Proc., AG, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 59–118 (2006)Google Scholar
17. 17.
Fripertinger, H., Reich, L.: The formal translation equation and formal cocycle equations for iteration groups of type I. Aequ. Math. 76(1–2), 54–91 (2008)Google Scholar
18. 18.
Fripertinger, H., Reich, L.: The formal translation equation for iteration groups of type II. Aequ. Math. 79(1–2), 111–156 (2010)Google Scholar
19. 19.
Knapp A.W.: Elliptic curves, Mathematical Notes, 40. Princeton University Press, Princeton (1992)Google Scholar
20. 20.
Lavrent’ev, M.A., Shabat, B.V.: Metody teorii funktsiĭ kompleksnogo peremennogo. (Russian) [Methods of the theory of functions in a complex variable] Fifth edition. Nauka, Moscow, 688 pp. (1987)Google Scholar
21. 21.
Moszner Z.: The translation equation and its application. Demonstr. Math. 6, 309–327 (1973)
22. 22.
Moszner Z.: General theory of the translation equation. Aequ. Math. 50(1-2), 17–37 (1995)
23. 23.
Nikolaev, I., Zhuzhoma, E.: Flows on 2-dimensional manifolds. An overview. Lecture Notes in Mathematics, 1705. Springer, Berlin (1999)Google Scholar
24. 24.
Schett A.: Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions. Math. Comp. 30, 143–147 (1976)
25. 25.
Viennot G.: Une interpretation combinatoire des coefficients des développments en série entiére des fonctions elliptiques de Jacobi. J. Combin. Theory Ser. B. 29, 121–133 (1980)