Absolute root separation

TL;DR

This work studies the absolute separation $\mathrm{abs\,sep}(P)$ of polynomials with integer coefficients, aiming to bound it from below in terms of degree $d$ and height $H(P)$. It develops general bounds via symmetric-function techniques, deriving three asymptotic regimes for pairs of roots (real-real, real-nonreal, and nonreal-nonreal) and obtaining exponents that improve prior results. The paper also conducts extensive low-degree experiments, using exhaustive searches and perturbation constructions, finding evidence that the true exponent is often $-(d+1)$ (notably for $d=3,4,5,6$) though not yet proven in general. In particular, the degree-3 case is shown to be optimal with a proven bound of $-4$, and explicit high-height families are constructed for degrees 4, 5, and 6 that realize $\mathrm{abs\,sep}\ll H(P)^{-(d+1)}$, suggesting that current algebraic approaches may be close to sharp but analytic methods may be needed for full resolution.

Abstract

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Proposition 1: Effective Version of the Fundamental Theorem of Symmetric Functions
• proof
• Proposition 2: Cauchy Bound
• proof
• Theorem 3
• Theorem 4
• Proposition 3
• proof