Baro, E.Vicente, J. deOtero, M.2023-06-172023-06-172018[1] Y. Abe. A statement of weierstrass on meromorphic functions which admit an algebraic addition theorem. J. Math. Soc. Japan, 57(3):709–723, 07 2005. [2] Y. Abe. Explicit representation of degenerate abelian functions and related topics. FJMS, 70(2):321–336, 11 2012. [3] E. Baro, J. de Vicente, and M. Otero. Locally C-Nash groups. ArXiv:1707.08171, 2017. [4] E. Baro, J. de Vicente, and M. Otero. Two-dimensional abelian locally K-Nash groups. ArXiv:1711.00806, 2017. [5] A. A. Belavin and V. G. Drinfel’d. Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funktsional. Anal. i Prilozhen., 16(3):1–29, 96, 1982. [6] S. Bochner. On the addition theorem for multiply periodic functions. Proceedings of the American Mathematical Society, 3(1):99–106, 1952. [7] S. Bochner and W.T. Martin. Several Complex Variables. Princeton Mathematical Series, vol. 10. Princeton University Press, Princeton, N. J., 1948. [8] R.C. Gunning and H. Rossi. Analytic functions of several complex variables. Prentice- Hall, Inc., Englewood Cliffs, N.J., 1965. [9] H. Hancock. Lectures on the theory of elliptic functions. Wiley, New York, NY, 1910. [10] J.J. Madden and C.M. Stanton. One-dimensional Nash groups. Pacific J. Math., 154(2):331–344, 1992. [11] P.J. Myrberg. Über systeme analytischer funktionen welche ein additions-theorem besitzen. Preisschriften gekrönt und heraussgegeben von der fürstlich Jablonowskischen Gesellschaft zu Leipzig, Teubner, Leipzig, 1922. [12] P. Painlevé. Leçons, sur la théorie analytique des équations différentielles: professées à Stockholm (septembre, octobre, novembre 1895) sur l’invitation de S. M. le roi de Suède et de Norwège. Number v. 11, no. 4. A. Hermann, 1894. [13] P. Painlevé. Sur les fonctions qui admettent un théorème d’addition. Acta Math., 27(1):1–54, 1903. [14] E. Phragmen. Sur un théorème concernant les fonctions elliptiques. Acta Math., 7(1):33–42, 1885. [15] J.M. Ruiz. The basic theory of power series. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1993. [16] F. Severi. Funzioni quasi abeliane. Pontificia Accademia delle Scienze Città del Vaticano: Pontificiae Academiae Scientiarum scripta varia. Pontificia Accademia delle scienze, 1961. [17] M. Shiota. Nash functions and manifolds. In Lectures in real geometry (Madrid, 1994), volume 23 of de Gruyter Exp. Math., pages 69–112. de Gruyter, Berlin, 1996. [18] M.B. Villarino. Algebraic additions theorems, preprint. Arxiv:1212.6471v2, 1998. [19] K. Weierstrass. Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen. Nach Vorlesungen und Aufzeichnungen des Herrn Professor K.Weierstrass. Bearbeitet und herausgegeben von H.A.Schwarz. Verlag von Julius Springer, Berlin, 1893.1559-002X10.1007/s12220-018-9992-7https://hdl.handle.net/20.500.14352/13160International Program of Excellence in Mathematics at Universidad Autónoma de MadridWe prove that if an analytic map : U [flecha] Cn, where U [incluye] C [elevado a ] n is an open neighborhood of the origin, admits an algebraic addition theorem then, there exists a meromorphic map g : C [elevado a]n [flecha] C [elevado a]n admitting an algebraic addition theorem such that each coordinate function of f is algebraic over C(g) on U (this was proved by K. Weierstrass for n = 1). Furthermore, g admits a rational addition theorem.engjournal articlehttps://link.springer.com/article/10.1007/s12220-018-9992-7open access512ÁlgebraAlgebraic addition theoremRational addition theoremNash groupsMatemáticas (Matemáticas)Álgebra12 Matemáticas1201 Álgebra