Georgia Institute of Technology College of Sciences

In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.