Circuits versus trees in algebraic complexity

Abstract

This survey is devoted to some aspects of the “P = NP ?” problem over the real numbers and more general algebraic structures. We argue that given a structure M, it is important to find out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees can be efficiently simulated by circuits. Point location, a problem of computational geometry, comes into play in the study of these questions for several structures of interest.