Discriminant and root separation of integral polynomials

Abstract

Consider a random polynomial $$G_Q(x)={\xi }_{Q,n}{x}^n+{\xi }_{Q,n-1}{x}^{n-1}+...+{\xi }_{Q,0}$$ 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 $$\underset{-\infty \le a\le b\le \infty }{\sup }|\mathbb{P}(a\le \frac{D(G_Q)}{Q^{2n-2}}\le b)-{\int }_a^b{\phi }_n(x)dx|\le \frac{C_n}{\log Q},$$ where ${\phi }_n$ 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 $\Delta (G_Q)$ denote the minimal distance between the complex roots of $G_Q$. As an application we show that for any $\epsilon >0$ 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 $$\mathbb{P}({\delta }_n<\Delta (G_Q)<\frac{1}{{\delta }_n})>1-\epsilon .$$

References in corpus (2)

• On the number of integral quadratic polynomials with bounded heights and discriminants
• Integral polynomials with small discriminants and resultants

Cited by in corpus (3)

• Integral polynomials with small discriminants and resultants
• Correlations between real and complex zeros of a random polynomial
• Distribution of complex algebraic numbers