The real zeros of a random algebraic polynomial with dependent coefficients

Abstract

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of 2 π log n, as n → ∞. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case looked at coefficients with an exponentially decaying covariance function, while the second assumed a constant covariance. He showed that the expectation of the number of real zeros for an exponentially decaying covariance matches the independent case, while having a constant covariance reduces the expected number of zeros in half. In this paper we will apply techniques similar to Sambandham's and extend his results to a wider class of covariance functions. Under certain restrictions on the spectral density, we will show that the order of the expected number of real zeros remains the same as in the independent case.

The -level crossings of a random algebraic polynomial with dependent coefficients

“…where now the expression on the right is the expected number of real zeros of P n (x) on (−∞, −1). Thus, Theorem 1.1 in [6] yields the upper bounds (3.5)…”

“…Notice that this upper bound holds on the entire interval (−1, 1). Next, from equations (3.5) and (3.7) in [6] we know that…”

“…Let g(y) = y log n log log n . Starting with x = 1−y ∈ (1− 1 log n , 1− log log n n ), from (3.2), (3.5), and (3.9) in [6] we have the equations…”

“…where f (φ) is the spectral density of the covariance function (in addition to the discussion in [6], see [1] and [2] for further references). By imposing certain conditions on the spectral density, for the random polynomial P n (x) given by (1.1), we will be able to study the level crossings for a wide range of covariance functions.…”

“…These results will be proved using the techniques developed by Farahmand in [3] and [4], as well as the spectral density of the covariance function. We will also make use of several results from [6], which in turn draws heavily from the work of Sambandham in [7]. Letting N K (α, β) be the number of K-level crossings of P n (x) in the interval (α, β), the main theorem is formulated as follows.…”