Graphs of zeros of analytic families

Author:
Alexander Brudnyi

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1859-1875

MSC (2000):
Primary 30C15; Secondary 34C07, 05C99

Published electronically:
September 23, 2004

MathSciNet review:
2115079

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a family of holomorphic functions in a domain depending holomorphically on . We study the distribution of zeros of in a subdomain whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.

**[B]**Alan Baker,*Transcendental number theory*, Cambridge University Press, London-New York, 1975. MR**0422171****[BM]**Edward Bierstone and Pierre D. Milman,*Semianalytic and subanalytic sets*, Inst. Hautes Études Sci. Publ. Math.**67**(1988), 5–42. MR**972342****[Br1]**Alexander Brudnyi,*On local behavior of analytic functions*, J. Funct. Anal.**169**(1999), no. 2, 481–493. MR**1730560**, 10.1006/jfan.1999.3481**[Br2]**Alexander Brudnyi,*Small amplitude limit cycles and the distribution of zeros of families of analytic functions*, Ann. of Math. (2)**154**(2001), no. 2, 227–243. MR**1865970**, 10.2307/3062096**[H]**Heisuke Hironaka,*Subanalytic sets*, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR**0377101****[Ha]**Robert M. Hardt,*Triangulation of subanalytic sets and proper light subanalytic maps*, Invent. Math.**38**(1976/77), no. 3, 207–217. MR**0454051****[I]**Yu. Ilyashenko,*Centennial history of Hilbert’s 16th problem*, Bull. Amer. Math. Soc. (N.S.)**39**(2002), no. 3, 301–354. MR**1898209**, 10.1090/S0273-0979-02-00946-1**[RY]**N. Roytwarf and Y. Yomdin,*Bernstein classes*, Ann. Inst. Fourier (Grenoble)**47**(1997), no. 3, 825–858 (English, with English and French summaries). MR**1465788**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30C15,
34C07,
05C99

Retrieve articles in all journals with MSC (2000): 30C15, 34C07, 05C99

Additional Information

**Alexander Brudnyi**

Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03615-3

Keywords:
Graph of zeros,
exponential polynomial,
displacement map

Received by editor(s):
August 7, 2003

Published electronically:
September 23, 2004

Additional Notes:
The author’s research was supported in part by NSERC

Article copyright:
© Copyright 2004
American Mathematical Society