On the lower bound of the number of real roots of a random algebraic equation
Let \(F_n(x,\omega)\equiv F(x)=\sum^n_{\nu=0} a_\nu(\omega)b_\nu x^\nu\), where \(\{a_\nu(\omega)\}^n_{\nu=0}\) is a sequence of normally distributed random variables defined on a probability space \((\Omega,{\mathcal A},\text{Pr})\) and \(\{b_\nu\}^n_{\nu=0}\) is a sequence of positive numbers. There are many results concerning the upper and lower bounds of the number of real zeros of \(F(x)\) under some assumption for the distribution of the coefficients \(a_\nu\) and some conditions for \(b_\nu\). The present paper obtains a lower bound for the number of real zeros for a case of dependent \(a_\nu\). It is assumed that the constants \(b_\nu\) satisfy the relation \(k_n/t_n= o(\log n)\), where \(k_n=\max_{0\leq\nu\leq n}b_\nu\) and \(t_n=\min_{0\leq\nu\leq n}b_\nu\). It is shown that for \(n\) large and \(C\) and \(C'\) positive constants, outside a set of measure at most \(C'\log((k_n/t_n)\log n)/\log n\) there are at least \(C\log n/\{\log((k_n/t_n)\log n)\}\) real zeros for \(F(x)\). The proof is similar to the earlier works of \textit{G. Samal} [Proc. Camb. Philos. Soc. 58, 433-442 (1962; Zbl 0113.34201)] and \textit{G. Samal} and \textit{M. N. Mishra} [Proc. Am. Math. Soc. 33, 523-528 (1972; Zbl 0214.16102) and ibid. 39, 184-189 (1973; Zbl 0274.60044)]. However, the details differ to accommodate the constants \(b_\nu\) and the dependent coefficients.
scientific article; zbMATH DE number 954944