Geodesic
The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. Application to solution of algebraic equations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 94-118 
L. S. Maergoiz. The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. Application to solution of algebraic equations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 94-118. http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a5/
@article{ZNSL_2020_491_a5,
     author = {L. S. Maergoiz},
     title = {The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. {Application} to solution of algebraic equations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {94--118},
     year = {2020},
     volume = {491},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a5/}
}
TY  - JOUR
AU  - L. S. Maergoiz
TI  - The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. Application to solution of algebraic equations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2020
SP  - 94
EP  - 118
VL  - 491
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a5/
LA  - ru
ID  - ZNSL_2020_491_a5
ER  - 
%0 Journal Article
%A L. S. Maergoiz
%T The multivalent indicator and conjugate diagrams of an entire function of order $\rho \neq 1$. Application to solution of algebraic equations
%J Zapiski Nauchnykh Seminarov POMI
%D 2020
%P 94-118
%V 491
%U http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a5/
%G ru
%F ZNSL_2020_491_a5

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper gives a survey of recent advances in the growth theory of entire functions associated with a theorem of G. Pólya describing the relationship between the indicator and conjugate diagrams for entire functions of exponential type. We discuss several methods of analytic continuation of a multivalued function of one variable given on a part of its Riemann surface in the form of a Puiseux series generated by the power function $z = w^{1/\rho}$, where $\rho > 1/2,\ \rho \neq 1$. We present a multivalent variant of the mentioned theorem of G. Pólya. The description is based on a geometrical construction of V. Bernstein for the multivalent indicator diagram of an entire function of order $\rho \neq 1$ and of normal type. We extend the method of E. Borel, which allows one to find the region of summability for a “proper” Puiseux series (the multivalent Borel polygon). This result seems to be new even in the case of power series. The theory applies to description of the domains of analytic continuation for Puiseux series representing the inverse functions for rational ones. As but one consequence we elaborate a new approach to solution of algebraic equations.

[1] G. Pólya, “Untersuchungen über Lücken and Singularitäten von Potenzreihen”, Math. Zeits., 29 (1929), 549–640 | DOI | MR | Zbl

[2] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographes, 150, Amer. Math. Soc., Providence, R.I., 1996 | DOI | MR | Zbl

[3] Л. С. Маергойз, Асимптотические характеристики целых функций и их приложения в математике и биофизике, Наука, Сиб. Отделение, Новосибирск, 1991 | MR | Zbl

[4] E. Borel, Leçons sur les Séries Divergentes, Gaunter-Villars, Paris, 1901 ; The second edition, Paris, 1928 | MR

[5] V. Bernstein, “Sulle proprieta caratteristiche delle indicatrici di crescenza delle transcendenti intere d'ordine finito”, Memoire della classe di scien. fis. mat. e natur., 6 (1936), 131–189

[6] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970 | MR | Zbl

[7] L. S. Maergoiz, “Ways of Analytic Continuation of Many-Valued Function of One Variable. Applications”, International conference “COMPLEX ANALYSIS AND ITS APPLICATIONS” dedicated to the 90th birth anniversary of I.P. Mityuk, Abstracts (Gelendzhik — Krasnodar, Russia 2018), 75 | MR | Zbl

[8] L. S. Maergoiz, “Analytic continuation methods for multivalued functions of one variable and their application to the solution of algebraic equations”, 28th St. Petersburg Summer Meeting in Mathematical Analysis, Abstracts, Euler International Mathematical Institute, 2019

[9] L. S. Maergoiz, “Mnogolistnye varianty teorem Poia–Bernshteina, Borelya dlya tselykh funktsii poryadka $\rho \neq 1$ i ikh prilozheniya”, Doklady PAN, 478:3 (2018), 266–270 | Zbl

[10] L. S. Maergoiz, “Cposoby analiticheskogo prodolzheniya mnogoznachnykh funktsii odnoi peremennoi. Prilozhenie k resheniyu algebraicheskikh uravnenii”, Trudy Inst. matem. i mekh. Ur.O RAN, 25, 2019, 120–135

[11] R. Nevanlinna, Uniformisierung, Berlin, 1953 | MR | Zbl

[12] J. Riordan, An Introduction to Combinatorial Analysis, Jorn Wiley and Sons, New York, 1958 | MR

[13] H. J. Mellin, “Résolution de l'équation Algébrique Générale à l'aide de la Fonction Gamma”, C. R. Acad. Sci., 172 (1921), 658–661 | Zbl

[14] T. M. Sadykov, A. K. Tsikh, Gipergeometricheskie i algebraicheskie funktsii mnogikh peremennykh, Nauka, M., 2014