z)?(x)?=??(?(xz)(1?em class="EmphasisTypeItalic">z)/z); here ?(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." />
The projective translation equation and unramified 2-dimensional flows with rational vector fields
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  • 作者：Giedrius Alkauskas
  • 关键词：Primary 39B12 ; 37E35 ; Secondary 14H52 ; 14H05 ; 14E05 ; 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
  • 刊名：Aequationes Mathematicae
  • 出版年：2015
  • 出版时间：June 2015
  • 年：2015
  • 卷：89
  • 期：3
  • 页码：873-913
  • 全文大小：937 KB
  • 参考文献：1.Aczél, J.: Lectures on functional equations and their applications. Mathematics in Science and Engineering, Vol. 19 Academic Press, New York-London (1966)
    2.Ahiezer, N.I.: èlementy teorii èllipti?eskih funkci?. (Russian) [Elements of the Theory of Elliptic Functions] Gosudarstv. Izdat. Tehn. Teor. Lit., Moscow-Leningrad,. 291?pp (1948)
    3.Alkauskas G.: Multi-variable translation equation which arises from homothety. Aequ. Math. 80(3), 335-50 (2010)View Article MATH MathSciNet
    4.Alkauskas G.: The projective translation equation and rational plane flows. I. Aequ. Math. 85(3), 273-28 (2013)View Article MATH MathSciNet
    5.Alkauskas, G.: The projective translation equation and rational plane flows. II (in preparation)
    6.Alkauskas, G.: The projective translation equation: rational vector fields and quasi-flows (tentative title)
    7.Bacher R., Flajolet Ph.: Pseudo-factorials, elliptic functions, and continued fractions. Ramanujan J. 21(1), 71-7 (2010)View Article MATH MathSciNet
    8.Blasiak, P., Flajolet, Ph.: Combinatorial models of creation-annihilation. Sém. Lothar. Combin. 65, Art. B65c (2011)
    9.Dixon, A.C.: On the doubly periodic functions arising out of the curve x 3 +? y 3 ?3αxy?= 1. Quart. J. XXIV, 167-33 (1890)
    10.Dumont D.: A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions. Math. Comp. 33, 1293-297 (1979)View Article MATH MathSciNet
    11.Dumont D.: Une approche combinatoire des fonctions elliptiques de Jacobi. Adv. Math. 1, 1-9 (1981)View Article MathSciNet
    12.Dumont, D.: Grammaires de William Chen et dérivations dans les arbres et arborescences. Sém. Lothar. Combin. 37, Art. B37a, 21?pp. (electronic) (1996)
    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)
    14.Flajolet Ph., Gabarró J., Pekari H.: Analytic urns. Ann. Probab. 33(3), 1200-233 (2005)View Article MATH MathSciNet
    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)
    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, 59-18 (2006)
    17.Fripertinger, H., Reich, L.: The formal translation equation and formal cocycle equations for iteration groups of type I. Aequ. Math. 76(1-), 54-1 (2008)
    18.Fripertinger, H., Reich, L.: The formal translation equation for iteration groups of type II. Aequ. Math. 79(1-), 111-56 (2010)
    19.Knapp A.W.: Elliptic curves, Mathematical Notes, 40. Princeton University Press, Princeton (1992)
    20.Lavrent’ev, M.A., Shabat, B.V.: Metody teorii funktsi? kompleksnogo peremennogo. (Russian) [Methods of the theory of functions in a complex variable] Fifth edition. Nauka, Moscow, 688?pp. (1987)
    21.Moszner Z.: The translation equation and its application. Demonstr. Math. 6, 309-27 (1973)MATH MathSciNet
    22.Moszner Z.: General theory of the translation equation. Aequ. Math. 50(1-2), 17-7 (1995)View Article MathSciNet
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    25.Viennot G.: Une interpretation combinatoire des coefficients des développments en série entiére des fonctions elliptiques de Jacobi. J. Combin. Theory Ser. B. 29, 121-33 (1980)View Article MATH MathSciNet
  • 作者单位：Giedrius Alkauskas (1)

    1. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Mathematics
    Analysis
    Combinatorics
  • 出版者：Birkh盲user Basel
  • ISSN：1420-8903
文摘
Let x?=?(x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1?em class="EmphasisTypeItalic">z)?(x)?=??(?(xz)(1?em class="EmphasisTypeItalic">z)/z); here ?(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.

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