Discriminant and root separation of integral polynomials

Abstract

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 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 $Q\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \sup_{-\infty\leq a\leq b\leq\infty}|\mathbb{P}(a\leq \frac{D(G_Q)}{Q^{2n-2}}\leq b)-\int_a^bφ_n(x)\, dx|\leq\frac{C_n}{\log Q}, $$ where $φ_n$ denotes the distribution function of the discriminant of a random polynomial of degree $n$ with independent coefficients which are uniformly distributed on $[-1,1]$. Let $Δ(G_Q)$ 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 $δ_n>0$ such that $Δ(G_Q)$ is stochastically bounded from below/above for all sufficiently large $Q$ in the following sense $$ \mathbb{P}(δ_n<Δ(G_Q)<\frac1{δ_n})>1-\varepsilon . $$