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性的基本度量，因为它衡量了分离其零点的困难程度。在正维情形下，可达半径的概念取代了分离界。本文给出了光滑代数簇可达半径的若干界，这些界涉及多个重要不变量：条件数、Smale的$\gamma$和比特尺寸。我们还在一些一般性假设下，为随机代数簇提供了概率意义上的界。