Zeros of random hyperbolic and random algebraic polynomials

✍ Scribed by P. Hannigan

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 215 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00431-x

✦ Synopsis

In this paper we find the expected number of zero crossings for a general algebraic polynomial, (\sum_{j=1}^{n} a_{j}\left(\alpha x^{j}+\beta x^{-j}\right)), and a general hyperbolic polynomial, (\sum_{j=1}^{n} a_{j}(\alpha \cosh j x+\beta \sinh j x)), where (\alpha) and (\beta) are constants, and (a_{1}, a_{2}, \ldots, a_{n}), is a sequence of independent, normally distributed random variables with mean zero and variance (\sigma^{2}(\neq 0)). The asymptotic results obtained are independent of the values of the constants (\alpha) and (\beta), and are consistent with the results in the literature for the more commonly studied algebraic polynomial, where (\beta=0), and the more frequently considered hyperbolic polynomials, where either (\alpha=0) or (\beta=0).