Ultrametric smale's α-theory

We consider a random polynomial system with m equations and m real unknowns. Assume all equations have the same degree d and the law on the coefficients satisfies the Kostlan-Shub-Smale hypotheses. It is known that E(N^X) = d^m/2 where N^X denotes the ...

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton's method. In this paper we reconsider his algorithm in the light of ...

Information processed by complex cognitive systems is characterized by the presence of various closely connected hierarchic structures. The most natural geometry for the representation of such structures is geometry of trees and the corresponding ...