A Remark on Holomorphic Functions Rational in Some Variables

Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remain...

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Published in:	Siberian mathematical journal Vol. 65; no. 5; pp. 1112 - 1115
Main Authors:	Ivanova, O. A., Melikhov, S. N.
Format:	Journal Article
Language:	English
Published:	Moscow Pleiades Publishing 01.09.2024 Springer Nature B.V
Subjects:	Analytic functions Complex variables Conflicts of interest Hankel matrices Mathematics Mathematics and Statistics Polynomials Rationality Variables Hankel matrix rational function holomorphic function 517.553
ISSN:	0037-4466, 1573-9260
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Abstract	Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remaining variables. We use the Kronecker method for establishing some criterion for the rationality of a holomorphic function of one complex variable in a neighborhood of zero by using the properties of Hankel matrices.
AbstractList	Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remaining variables. We use the Kronecker method for establishing some criterion for the rationality of a holomorphic function of one complex variable in a neighborhood of zero by using the properties of Hankel matrices. Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other variables, we prove that the functions are representable as a ratio of polynomials in these variables whose coefficients are holomorphic in the remaining variables. We use the Kronecker method for establishing some criterion for the rationality of a holomorphic function of one complex variable in a neighborhood of zero by using the properties of Hankel matrices.
Author	Melikhov, S. N. Ivanova, O. A.
Author_xml	– sequence: 1 givenname: O. A. surname: Ivanova fullname: Ivanova, O. A. email: neo_ivolga@mail.ru organization: Vorovich Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University – sequence: 2 givenname: S. N. orcidid: 0000-0002-5895-9607 surname: Melikhov fullname: Melikhov, S. N. organization: Vorovich Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University, Southern Mathematical Institute
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Cites_doi	10.4064/ap-12-2-139-142
ContentType	Journal Article
Copyright	Pleiades Publishing, Ltd. 2024. Russian Text © The Author(s), 2024, published in Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 5, pp. 921–925. Pleiades Publishing, Ltd. 2024.
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References	PolyaGSzegöGProblems and Theorems in Analysis. II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry1997BerlinSpringerClassics in Mathematics RudinWStoutELBoundary properties of functions of several complex variablesJ. Math. Mech.1965149911005182748 IvanovPAMelikhovSNPommiez operator in spaces of analytic functions of several complex variablesJ. Math. Sci.20212523345359 Kronecker L., “Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen,” Monatsber. Königl. Preuss. Akad. Wiss. (Berlin), 535–600 (1881). ShabatBVAn Introduction to Complex Analysis. Part 2: Functions of Several Variables1992ProvidenceAmer. Math. Soc. SiciakJA note on rational functions of several complex variablesAnn. Polon. Math.19621213914214766910.4064/ap-12-2-139-142 W Rudin (1479_CR6) 1965; 14 G Polya (1479_CR2) 1997 J Siciak (1479_CR3) 1962; 12 BV Shabat (1479_CR5) 1992 PA Ivanov (1479_CR4) 2021; 252 1479_CR1
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Snippet	Considering the holomorphic functions in a polycylindrical domain which are rational in some of the variables for the arbitrary fixed values of other...
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SubjectTerms	Analytic functions Complex variables Conflicts of interest Hankel matrices Mathematics Mathematics and Statistics Polynomials Rationality Variables
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Title	A Remark on Holomorphic Functions Rational in Some Variables
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