Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS160, part 1: Numerical methods for structured polynomial system solving
Time:
Tuesday, 09/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F012
30 seats, 57m^2

Presentations
3:00pm - 5:00pm

Numerical methods for structured polynomial system solving

Chair(s): Alperen Ergur (TU Berlin), Pierre Lairez (INRIA), Gregorio Malajovich (Universidade Federal do Rio de Janeiro, Brazil), Josue Tonelli-Cueto (TU Berlin)

Improvements in the understanding of numerical methods for dense polynomial system solving led to the complete solution of Smale's 17th problem. At this point, it remains an open challenge to achieve the same success in the solution of structured polynomial systems: explain the typical behavior of current algorithms and devise polynomial-time algorithms for computing roots of polynomial systems. In this minisymposium, researchers will present the current progress on applying numerical methods to structured polynomial systems.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Introductory Talk

Alperen Ergur
TU Berlin

This talk will provide a brief overview of the state of the art in numerical methods for polynomial system solving, and introduce the goals of the mini-symposium.

 

On the condition number of some algebraic problems.

Diego Armentano1, Carlos Beltrán2
1Universidad de la Republica, 2Universidad de Cantábria

In this talk we will introduce a geometric framework to analyze the condition number of some classic algebraic problems. We will compute the average value of the squared condition number, when the input is taken at random, for a large family of these problems. In particular, we will show that the polynomial eigenvalue problem is quite well conditioned.

 

Numerical irreducible decomposition with one homotopy

Dan Bates1, David Eklund2, Jonathan Hauenstein3, Chris Peterson4
1US Naval Academy, 2KTH, 3University of Notre Dame, 4Colorado State University

A numerical irreducible decomposition (NID) of an algebraic set V includes a set of witness points (approximations of generic points) on each irreducible component of V, along with various auxiliary data. The computation of an NID typically involves a sequence of homotopies. Pairing together the machinery of excess intersection and isosingular sets, we show how to compute an NID with only one homotopy. This is joint work with David Eklund, Jonathan Hauenstein, and Chris Peterson.

 

Computing the Homology of arbitrary Semialgebraic Sets

Felipe Cucker1, Peter Bürgisser2, Josue Tonelli-Cueto2
1City University of Hong Kong, 2TU Berlin

We describe recent advances regarding the computation of the homology groups of arbitrary semialgebraic sets. These advances follow the line of results obtained for the particular cases of smooth projective varieties and basic semialgebraic sets.

Coauthors are Peter Buergisser and Josue Tonelli-Cueto.