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Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle.
- Source :
-
Transactions of the American Mathematical Society . Dec1999, Vol. 351 Issue 12, p4963-4980. 18p. - Publication Year :
- 1999
-
Abstract
- Let $T(x)=\sum_{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min_{x\in[0,2\pi)}|T(x)|<n^{-1/2+\epsilon}$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon>0$ and large $n$ most choices of sign satisfy $\min_{x\in[0,2\pi)}|T(x)|> \eps n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials. [ABSTRACT FROM AUTHOR]
- Subjects :
- *PROBABILITY theory
*RANDOM polynomials
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 351
- Issue :
- 12
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 9596344
- Full Text :
- https://doi.org/10.1090/S0002-9947-99-02241-2