The D-plus Discriminant and Complexity of Root Clustering

Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $\alpha_1,\ldots,\alpha_m$ whose multiplicities are $\boldsymbol{\mu}=(\mu_1,\ldots,\mu_m)$. We define the D-plus discriminant of $p(x)$ to be $D^+(p):= \prod_{1\le i<j\le m}(\alpha_i-\alpha_j)^{\mu_i+\mu_j}$. Unlike the classical discriminant, $D^+(p)$ never vanishes. We first prove a conjecture that $D^+(p)$ is a $\boldsymbol{\mu}$-symmetric function of its roots $\alpha_1,\ldots,\alpha_m$. Our main result gives an explicit formula for $D^+(p)$, as a rational function of its coefficients. A basic tool used by our proof is the "symbolic Poisson resultant". 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.