Discriminant and root separation of integral polynomials

F. Götze ; D. Zaporozhets

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 144-153

Résumé

Consider a random polynomial $$ G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+\dots+\xi_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q,\dots,Q\}$. 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 $Q\ge2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \sup_{-\infty\leq a\leq b\leq\infty}\left|\mathbf P\left(a\leq\frac{D(G_Q)}{Q^{2n-2}}\leq b\right)-\int_a^b\varphi_n(x)\,dx\right|\leq\frac{C_n}{\log Q}, $$ where $\varphi_n$ denotes the probability density function of the discriminant of a random polynomial of degree $n$ with independent coefficients which are uniformly distributed on $[-1,1]$. Let $\Delta(G_Q)$ denote the minimal distance between the complex roots of $G_Q$. As an application we show that for any $\varepsilon>0$ there exists a constant $\delta_n>0$ such that $\Delta(G_Q)$ is stochastically bounded from below/above for all sufficiently large $Q$ in the following sense $$ \mathbf P\left(\delta_n\Delta(G_Q)\frac1{\delta_n}\right)>1-\varepsilon. $$