Recent Advances in Real Complexity and Computation
About this Title
José Luis Montaña, Ciencias Universidad de Cantabria, Santander, Spain and Luis M. Pardo, Universidad de Cantabria, Santander, Spain, Editors
Publication: Contemporary Mathematics
Publication Year 2013: Volume 604
ISBNs: 978-0-8218-9150-6 (print); 978-1-4704-1409-2 (online)
This volume is composed of six contributions derived from the lectures given during the UIMP-RSME Lluís Santaló Summer School on “Recent Advances in Real Complexity and Computation”, held July 16–20, 2012, in Santander, Spain.
The goal of this Summer School was to present some of the recent advances on Smale's 17th Problem: “Can a zero of $n$ complex polynomial equations in $n$ unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?”
These papers cover several aspects of this problem: from numerical to symbolic methods in polynomial equation solving, computational complexity aspects (both worse and average cases and both upper and lower complexity bounds) as well as aspects of the underlying geometry of the problem. Some of the contributions also deal with either real or multiple solutions solving.
Graduate students and research mathematicians interested in the complexity of computation.
Table of Contents
- Martijn Baartse and Klaus Meer – Topics in real and complex number complexity theory
- Bernd Bank, Marc Giusti and Joos Heintz – Polar, bipolar and copolar varieties: Real solving of algebraic varieties with intrinsic complexity
- Carlos Beltrán and Michael Shub – The complexity and geometry of numerically solving polynomial systems.
- M. Giusti and J.-C. Yakoubsohn – Multiplicity hunting and approximating multiple roots of polynomial systems
- Joos Heintz, Bart Kuijpers and Andrés Rojas Paredes – On the intrinsic complexity of elimination problems in effective algebraic geometry
- Gregorio Malajovich – Newton iteration, conditioning and zero counting