Discriminant and Root Separation of Integral Polynomials

Abstract

Consider a random polynomial G_Q(x) = ξ_Q,nxⁿ + ξ_{Q,n − 1}x^n − 1 + ⋯ + ξ_Q,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, . . .,Q}. Denote by D(G_Q) the discriminant of G_Q. We show that there exists a constant C_n depending on n only such that for all Q ≥ 2, the distribution of D(G_Q) can be approximated as follows: \( \underset{-\infty \le a\le b\le -\infty }{ \sup}\left|\mathrm{P}\left(a\frac{D\left({G}_Q\right)}{Q^{2n-2}}\le b\right)-{\displaystyle \underset{a}{\overset{b}{\int }}{\upvarphi}_n(x)dx}\right|\le \frac{C_n}{ \log Q}, \) where \( \varphi \)_n denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(G_Q) denote the minimal distance between complex roots of G_Q. As an application, we show that for any ε > 0 there exists a constant δ_n > 0 such that Δ(G_Q) is stochastically bounded from below/above for all sufficiently large Q in the following sense: \( \mathrm{P}\left({\delta}_n<\varDelta \left({G}_Q\right)<\frac{1}{\delta_n}\right)>1-\varepsilon \). Bibliography: 14 titles.