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- W2950891880 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 $Qge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ sup_{-inftyleq aleq bleqinfty}|mathbb{P}(aleq frac{D(G_Q)}{Q^{2n-2}}leq b)-int_a^bvarphi_n(x), dx|leqfrac{C_n}{log Q}, $$ where $varphi_n$ 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 $varepsilon>0$ 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 $$ mathbb{P}(delta_n 1-varepsilon . $$" @default.
- W2950891880 created "2019-06-27" @default.
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- W2950891880 date "2014-07-23" @default.
- W2950891880 modified "2023-09-27" @default.
- W2950891880 title "Discriminant and root separation of integral polynomials" @default.
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