Complexity of Symbolic and Numerical Problems

Summary

The seminar was dedicated to Prof. Dima Grigoriev on the occasion of his 60th birthday. Its aim was to discuss modern trends in computational real algebraic geometry, in particular, areas related to solving real algebraic and analytic equations and inequalities. Very recent new developments in the analysis of these questions from the point of view of tropical mathematics were also presented.

Historically there were two strands in the computational approach to polynomial systems' solving. One is the tradition of numerical analysis, a classical achievement of which is the Newton's method. Various other approximation algorithms were developed since then, some based on the idea of a homotopy. Numerical analysis did not bother to introduce formal models of computations (and hence computational complexity considerations) but developed refined methods of estimations of convergency rates. Another tradition emerged from algebra, particularly in classical works of Cayley, Sylvester and Macaulay. Algebraic results concerning real solutions go further back to the Descartes' rule and Sturm sequences. An important contribution to the subject from logic was Tarski's constructive quantifier elimination procedures for algebraically closed and real closed fields. The computations considered in this tradition are exact, under modern terminology - "symbolic". They naturally fit into standard models of computation (Turing Machines, straight-line programs, computation trees) thus lending themselves to complexity analysis.

Until 1990s these two strands developed largely independently. One of the important unifying ideas became the concept of a real numbers (or BSS) machine suggested by Blum, Shub and Smale which can be considered as a model of computation for the numerical analysis. This idea led to Smale's 9th and 17th problems, which became an inspiration for many researchers in the field.

The seminar considered a wide set of questions related to the current state of the symbolic and numeric approaches to algorithmic problems of real algebraic and analytic geometry, also from the novel perspective of tropical and max/plus mathematics.