http://www.dagstuhl.de/15242

June 7 – 12 , 2015, Dagstuhl Seminar 15242

Complexity of Symbolic and Numerical Problems

Organizers

Peter Bürgisser (TU Berlin, DE)
Felipe Cucker (City University – Hong Kong, HK)
Marek Karpinski (Universität Bonn, DE)
Nicolai Vorobjov (University of Bath, GB)

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Dagstuhl Service Team

Documents

Dagstuhl Report, Volume 5, Issue 6 Dagstuhl Report
Aims & Scope
List of Participants
Shared Documents
Dagstuhl Seminar Schedule [pdf]

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.

License
  Creative Commons BY 3.0 Unported license
  Peter Bürgisser, Felipe Cucker, Marek Karpinski, and Nicolai Vorobjov

Classification

  • Data Structures / Algorithms / Complexity

Keywords

  • Symbolic computation
  • Algorithms in real algebraic geometry
  • Complexity lower bounds
  • Geometry of numerical algorithms

Book exhibition

Books from the participants of the current Seminar 

Book exhibition in the library, ground floor, during the seminar week.

Documentation

In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. The seminar organizers, in cooperation with the collector, prepare a report that includes contributions from the participants' talks together with a summary of the seminar.

 

Download overview leaflet (PDF).

Publications

Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.

Dagstuhl's Impact

Please inform us when a publication was published as a result from your seminar. These publications are listed in the category Dagstuhl's Impact and are presented on a special shelf on the ground floor of the library.

NSF young researcher support