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A probabilistic proof of the fundamental theorem of algebra
Author(s):
Mihai
N.
Pascu
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1707-1711.
MSC (2000):
Primary 30C15;
Secondary 60J65
Posted:
December 6, 2004
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Abstract:
We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.
References:
-
- 1.
- L. V. Ahlfors, Complex Analysis, McGraw-Hill, New York, third edition (1978). MR 0510197 (80c:30001)
- 2.
- R. Bass, Probabilistic Techniques in Analysis, Springer, New York (1995). MR 1329542 (96e:60001)
- 3.
- R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA (1984). MR 0750829 (87a:60054)
- 4.
- M. D. O'Neill, A geometric proof of the twist point theorem, Preprint (available at http://math.mckenna.edu/moneill).
- 5.
- M. N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), no. 11, pp. 4681 - 4702. MR 1926894 (2003i:60141)
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Additional Information:
Mihai
N.
Pascu
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
Address at time of publication:
Faculty of Mathematics and Computer Science, ``Transilvania'' University of Brasov, Str. Iuliu Maniu Nr. 50, Brasov, Jud. Brasov -- COD 2200, Romania
Email:
pascu@math.purdue.edu, mihai.pascu@unitbv.ro
DOI:
10.1090/S0002-9939-04-07700-7
PII:
S 0002-9939(04)07700-7
Keywords:
Brownian motion,
L\'{e}vy's theorem,
support theorem
Received by editor(s):
October 10, 2003
Received by editor(s) in revised form:
February 4, 2004
Posted:
December 6, 2004
Additional Notes:
This work was supported in part by NSF grant # 0203961 - DMS
Dedicated:
I dedicate this paper to my dear friend M. K.
Communicated by:
Richard C. Bradley
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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