The D-plus Discriminant and Complexity of Root Clustering

TL;DR

This paper introduces the D-plus discriminant for integer polynomials, proves its symmetry property, provides an explicit formula, and applies it to analyze the complexity of a root clustering algorithm.

Contribution

It proves the conjecture that the D-plus discriminant is symmetric and derives an explicit formula, connecting algebraic properties to algorithmic complexity.

Findings

D-plus discriminant is symmetric in roots.

Explicit formula for D-plus discriminant as a rational function.

Upper bound on the complexity measure related to D-plus discriminant.

Abstract

Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $\rho_1,\ldots,\rho_m$ whose multiplicities are $\boldsymbol{\mu}=(\mu_1,\ldots,\mu_m)$. We define the D-plus discriminant of $p(x)$ to be $D^+(p):= \prod_{1\le i<j\le m}(\rho_i-\rho_j)^{\mu_i+\mu_j}$. We first prove a conjecture that $D^+(p)$ is a $\boldsymbol{\mu}$-symmetric function of its roots $\rho_1,\ldots,\rho_m$. Our main result gives an explicit formula for $D^+(p)$, as a rational function of its coefficients. Our proof is ideal-theoretic, based on re-casting the classic Poisson resultant as the "symbolic Poisson formula". The D-plus discriminant first arose in the complexity analysis of a root clustering algorithm from Becker et al. (ISSAC 2016). The bit-complexity of this algorithm is proportional to a quantity $\log(|D^+(p)|^{-1})$. As an application of our main result, we give an explicit upper bound on this quantity in terms of the degree of $p$ and its leading coefficient.