Journal of Mathematical Sciences

, Volume 219, Issue 5, pp 700–706

Discriminant and Root Separation of Integral Polynomials

Article

Consider a random polynomial G Q (x) = ξ Q,n x n  + 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(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) 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(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) 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.

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