Separation bounds are a fundamental measure of the complexity of solving a zero-dimensional system as it measures how difficult it is to separate its zeroes. In the positive dimensional case, the notion of reach takes its place. In this paper, we provide bounds on the reach of a smooth algebraic variety in terms of several invariants of interest: the condition number, Smale's $\gamma$ and the bit-size. We also provide probabilistic bounds for random algebraic varieties under some general assumptions.

翻译：分离界是衡量零维系统求解复杂度的基本度量，因为它刻画了分离系统零点的困难程度。在正维情形下，到达度的概念取而代之。本文基于若干关键不变量——包括条件数、Smale γ常数与比特长度——为光滑代数簇的到达度提供了界。同时，在一般假设条件下，我们给出了随机代数簇的概率性界。