[Paper Review] Discriminant and root separation of integral polynomials

Friedrich Götze, Dmitry Zaporozhets|arXiv (Cornell University)|Jul 23, 2014

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper studies the distribution of discriminants and root separations for random integral polynomials with coefficients uniformly chosen from $\{-Q, \dots, Q\}$. It establishes that as $Q \to \infty$, the normalized discriminant $D(G_Q)/Q^{2n-2}$ converges in distribution to the discriminant of a random polynomial with coefficients in $[-1,1]$, with error bounded by $C_n / \log Q$. This implies typical integral polynomials have root separation bounded away from zero and infinity, resolving a probabilistic model of root spacing.

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 $$ \mathbb{P}(δ_n1-\varepsilon . $$

Motivation & Objective

• To understand the typical behavior of discriminants and root separations in random integral polynomials with bounded coefficients.
• To determine the limiting distribution of the discriminant when coefficients are uniformly distributed over $\{-Q, \dots, Q\}$ as $Q \to \infty$.
• To establish stochastic lower and upper bounds on the minimal root separation $\Delta(G_Q)$ for large $Q$, showing it is stochastically bounded away from zero and infinity.
• To extend the analysis to resultants of random polynomial pairs, showing analogous convergence in distribution.

Proposed method

• Model random integral polynomials $G_Q$ with i.i.d. coefficients uniformly distributed on $\{-Q, \dots, Q\}$, and normalize them by $Q$ to obtain $G_Q/Q$.
• Use moment convergence and characteristic function techniques to show that the $k$-th moments of the normalized discriminant $D(G_Q/Q)$ converge to those of the discriminant of a random polynomial with coefficients in $[-1,1]$.
• Establish uniform error bounds on the distributional approximation using Esseen’s inequality and characteristic function comparison.
• Prove that the difference between the cumulative distribution functions of $D(G_Q)/Q^{2n-2}$ and the limiting distribution $\varphi_n(x)$ is bounded by $C_n / \log Q$.
• Apply the result to derive stochastic bounds on root separation $\Delta(G_Q)$, showing it is with high probability bounded away from zero and infinity.
• Extend the method to resultants of two independent random polynomials, proving similar convergence in distribution for $R(G_Q, F_Q)/Q^{m+n}$.

Experimental results

Research questions

• RQ1 How does the discriminant of a random integral polynomial with coefficients in $\{-Q, \dots, Q\}$ behave asymptotically as $Q \to \infty$?
• RQ2 Can the distribution of the normalized discriminant $D(G_Q)/Q^{2n-2}$ be approximated by the distribution of the discriminant of a polynomial with coefficients in $[-1,1]$?
• RQ3 What is the typical minimal root separation $\Delta(G_Q)$ for such random polynomials, and is it stochastically bounded away from zero and infinity?
• RQ4 How does the distribution of the resultant of two independent random integral polynomials behave under normalization?
• RQ5 What is the rate of convergence of the distribution of normalized discriminants to the limiting law?

Key findings

• The distribution of $D(G_Q)/Q^{2n-2}$ converges to the distribution $\varphi_n(x)$ of the discriminant of a random polynomial with coefficients in $[-1,1]$, with error bounded by $C_n / \log Q$ for some constant $C_n$ depending only on $n$.
• For any $\varepsilon > 0$, there exists $\delta_n > 0$ such that $\mathbb{P}(\delta_n < \Delta(G_Q) < 1/\delta_n) > 1 - \varepsilon$ for all sufficiently large $Q$, implying typical root separation is bounded away from zero and infinity.
• The minimal root separation $\Delta(G_Q)$ is stochastically bounded from below and above, meaning it does not tend to zero or infinity in probability as $Q \to \infty$.
• The normalized discriminant $D(G_Q)/Q^{2n-2}$ converges in distribution to the discriminant of a limiting random polynomial with coefficients in $[-1,1]$, with explicit error control.
• The same convergence result holds for the resultant $R(G_Q, F_Q)/Q^{m+n}$ of two independent random polynomials of degrees $n$ and $m$, respectively.
• The convergence rate is $O(1/\log Q)$, which is optimal for this class of coefficient distributions.