The growth of solutions of algebraic differential equations
Hayman, Walter K.
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 7 (1996), p. 67-73 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

Suppose that f⁢z is a meromorphic or entire function satisfying P⁢z,f,f′,…,fn=0 where P is a polynomial in all its arguments. Is there a limitation on the growth of f, as measured by its characteristic T⁢r,f? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.

Sia f⁢z una funzione meromorfa o intera dell'equazione P⁢z,f,f′,…,fn=0, dove P è un polinomio in tutti i suoi termini. Esiste una limitazione della crescita di f, considerata rispetto alla sua caratteristica T⁢r,f? La risposta a tale questione non è in generale nota. L'autore e i Teoremi Gol'dberg e Steinmetz danno una risposta positiva in alcuni casi. Vengono anche forniti alcuni esempi.

Publié le : 1996-10-01
@article{RLIN_1996_9_7_2_67_0,
     author = {Walter K. Hayman},
     title = {The growth of solutions of algebraic differential equations},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {7},
     year = {1996},
     pages = {67-73},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1996_9_7_2_67_0}
}

Hayman, Walter K. The growth of solutions of algebraic differential equations. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 7 (1996) pp. 67-73. http://gdmltest.u-ga.fr/item/RLIN_1996_9_7_2_67_0/

[1] Bank, S. B., On determining the growth of meromorphic solutions of algebraic differential equations having arbitrary entire coefficients. Nagoya Math. J., 49, 1973, 53-65. | MR 320400 | Zbl 0268.34010

[2] Bank, S. B., Some results on Analytic and Meromorphic Solutions of Algebraic Differential Equations. Advances in Mathematics, 15, 1975, 41-61. | MR 379940 | Zbl 0296.34005

[3] Bank, S. B., On the existence of meromorphic solutions of differential equations having arbitrarily rapid growth. J. reine angew. Math., 288, 1976, 176-182. | MR 430371 | Zbl 0337.34007

[4] Basu, N. - Bose, S. - Vijayaraghavan, T., A simple example for a Theorem of Vijayaraghavan. J. London Math. Soc., 12, 1937, 250-252. | JFM 63.0419.02 | MR 95760

[5] Borel, E., Mémoire sur les séries divergentes. Ann. Sci. École Norm. Sup., 16, 1899, 9-136. | JFM 30.0230.03

[6] Gol'Dberg, A. A., On single valued solutions of first order differential equations. Ukrain Mat. Zh., 8, 1956, 254-261 (Russian). | MR 85396 | Zbl 0072.09202

[7] Hayman, W. K., The local growth of power series: A survey of the Wiman-Valiron method. Canad. Math. Bull., 17 (3), 1974, 317-358. | MR 385095 | Zbl 0314.30021

[8] Laine, I., Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin-New York1993. | MR 1207139 | Zbl 0784.30002

[9] Steinmetz, N., Über das Anwachsen der Lösungen homogener algebraischer Differentialgleichungen zweiter Ordnung. Manuscripta Math., 32, 1980, 303-308. | MR 595424 | Zbl 0444.34035

[10] Steinmetz, N., Über die eindeutigen Lösungen einer homogenen algebraischer Differentialgleichung zweiter Ordnung. Ann. Acad. Sci. Fenn., AI7, 1982, 177-188. | MR 686638 | Zbl 0565.34005

[11] Valiron, G., Fonctions Analytiques. Presses Universitaires de France, Paris1954. | MR 61658 | Zbl 0055.06702

[12] Vijayaraghavan, T., Sur la croissance des fonctions définies par les équations différentielles. C. R. Acad. Sci., Paris, 194, 1932, 827-829. | Zbl 0004.00803

[13] Wittich, H., Neuere Untersuchungen über eindeutige analytische Funktionen. Springer, Berlin-Göttingen-Heidelberg1955. | MR 77620 | Zbl 0159.10103