Strong result for real zeros of random algebraic polynomials (Q5960921)

There are many known estimates of the number of real zeros \(N_n(\omega)\) for a random algebraic polynomial of the form \(F_n(x,\omega)= \sum^n_{\nu= 0} a_\nu b_\nu x^\nu\), selected at random, in which \(a_\nu(\omega)\)'s, \(\omega\in\Omega\), are random variables defined on a fixed probability space \((\Omega,{\mathcal A},\text{Pr})\) assuming real values only and \(b_\nu\)'s are non-zero ral numbers. For \(b_\nu\equiv 1\), \(\nu= 0,1,2,\dots, n\), and \(a_\nu(\omega)\)'s independent normally distributed with mathematical expectation zero and variance one, \textit{E. A. Evans} [Proc. Lond. Math. Soc., III. Ser. 15, 731-749 (1965; Zbl 0134.35703)] obtained a ``strong result'' for both upper bound and lower bound of \(N_n(\omega)\). \textit{G. Samal} [Proc. Camb. Philos. Soc. 58, 433-442 (1962; Zbl 0113.34201)] relaxed the normality assumptions and assumed that the coefficients \(a_\nu(\omega)\) have expectation zero and finite and non-zero variance and third absolute moments.NEWLINENEWLINENEWLINEThe present paper, by generalizing Samal's method, studies the case of dependent coefficients in which the random variables \(a_\nu(\omega)\)'s are normally distributed with mean zero and joint density function defined by \(|M|^{1/2}(2\pi)^{-(n+ 1)/2}\exp\{-(1/2){\mathbf a}' M{\mathbf a}\}\), where \(M^{-1}\) is the moment matrix with \(\rho_{ij}= 1\) for \(i=j\) and \(\rho_{ij}= \rho_{|i-j|}\) for \(1\leq|i-j|\leq m\) and \(\rho_{ij}= 0\) for \(|i-j|> m\) for a positive integer \(m\). As usual \({\mathbf a}\) is the column vector whose transpose is \({\mathbf a}'= (a_0(\omega), a_1(\omega),\dots, a_n(\omega))\). It is assumed \(0\leq \rho_j< 1\), \(j= 1,2,\dots, m\), and \(b_\nu\) are positive numbers such that \(\log(\kappa_n/t_n)= o(\log n)\), where \(\kappa_n= \max_{0\leq\nu\leq n}b_n\) and \(t_n= \min_{0\leq\nu\leq n} b_n\). The author proves that there exists an integer \(n_0\) sufficiently large such that for each \(n> n_0\) and outside an exceptional set \(N_n(\omega)\) is at least \(C\log n/\log((\kappa_n/t_n)\log n)\). The measure of the exceptional set does not exceed \(C'/\log\{\log n_0/\log((\kappa_{n_0}/t_{n_0})\log n_0)\}\) where \(C\) and \(C'\) are positive absolute constants. This result is strong in the sense that the measure of the exceptional set is independent of \(n\).