A probabilistic proof of the fundamental theorem of algebra
HTML articles powered by AMS MathViewer
- by Mihai N. Pascu PDF
- Proc. Amer. Math. Soc. 133 (2005), 1707-1711 Request permission
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
- Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
- Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
- Richard Durrett, Brownian motion and martingales in analysis, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1984. MR 750829
- M. D. O’Neill, A geometric proof of the twist point theorem, Preprint (available at http://math.mckenna.edu/moneill).
- Mihai N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4681–4702. MR 1926894, DOI 10.1090/S0002-9947-02-03020-9
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 Braşov, Str. Iuliu Maniu Nr. 50, Braşov, Jud. Braşov – COD 2200, Romania
- Email: pascu@math.purdue.edu, mihai.pascu@unitbv.ro
- Received by editor(s): October 10, 2003
- Received by editor(s) in revised form: February 4, 2004
- Published electronically: December 6, 2004
- Additional Notes: This work was supported in part by NSF grant # 0203961 - DMS
- Communicated by: Richard C. Bradley
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 1707-1711
- MSC (2000): Primary 30C15; Secondary 60J65
- DOI: https://doi.org/10.1090/S0002-9939-04-07700-7
- MathSciNet review: 2120250
Dedicated: I dedicate this paper to my dear friend M. K.