27 research outputs found
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
Computability of Julia sets
In this paper we settle most of the open questions on algorithmic
computability of Julia sets. In particular, we present an algorithm for
constructing quadratics whose Julia sets are uncomputable. We also show that a
filled Julia set of a polynomial is always computable.Comment: Revised. To appear in Moscow Math. Journa