Jan Verschelde, University Illinois Chicago, Chicago, IL
An Overview of Numerical Algebraic Geometry

An important emerging trend in computer algebra during recent years concerns the development of hybrid symbolic-numeric algorithms to solve problems with approximate (or empirical) input coefficients. Homotopy continuation methods lead to robust solvers of polynomials systems that are used in the tools of numerical algebraic geometry. While these tools can be viewed as hybrid symbolic-numeric algorithms, answers to problems of symbolic-numeric computing also stimulate the development of numerical algebraic geometry.


Zhonggang Zeng, Northeastern Illinois University, Chicago, IL
Numerical Computation of the Polynomial Irreducible Factorization

Computing the irreducible factorization of a multivariate polynomial in exact sense is an ill-posed problem. A polynomial generally loses its factorability if its coefficients are approximately given due to data measurement or round-off error in numerical computation. Computing such approximate factorization has been a problem of practical interest and several algorithms have been proposed with various features. In this talk we introduce a well-posed formulation of the approximate irreducible factorization and a two-staged new algorithm for its computation. The first stage of the algorithm identifies the factorization structure using matrix rank-revealing, eigendecompositions and polynomial approximate GCD computation. In the second stage the nearest polynomial subject to the constraint of the factorization structure is iteratively calculated along with the irreducible factorization. Preliminary implementation and numerical results will also be presented.


Gregory Reid, University of Western Ontario, London, ON
New Extensions and Applications for Numerical Algebraic Geometry

This talk will discuss some extensions, future directions and applications of Numerical Algebraic Geometry:

• Extension to systems of PDE. This is the new area of Numerical Jet Geometry.
• Applications to modeling using general PDE systems
• The role of genericity in numerical algebraic and jet geometry
• Open problems

Daniel Bates, Institute for Mathematics and its Applications (IMA), Minneapolis, MN
Numerical Algebraic Geometry in Control Theory

Certain problems arising in control theory may be addressed by repeatedly solving polynomial systems at various points in a parameter space. In recent years, the standard approach has been to solve each polynomial system with little or no knowledge from previous parameter values. For those aware of continuation methods, there is clearly a different solution. This talk will report on joint work with Ioannis Fotiou and Philipp Rostalski (both at ETH Zurich) on solving such control problems via continuation and on the potential for using more sophisticated methods from numerical algebraic geometry to produce more efficient techniques.


Barry Dayton, Northeastern Illinois University, Chicago, IL
Local Solution of Analytic Systems by Homotopy

We discuss the homotopy continuation approach to Rouche's theorem due to Verschelde and Haegemans (1994). One can deduce from this theorem that, under mild conditions, it is theoretically possible to find an polynomial system which serves as a homotopy continuation start system for a straight line homotopy finding all zeros of a holomorphic system within a distance r of some point. In practice it is difficult to find a system, and/or to verify the hypotheses for such a system. Motivated by an extension of Rouche's theorem it is, however, possible to construct certain random polynomial start systems that work quite well much of the time. Based on this we propose a black box algorithm to solve holomorphic systems in small dimensions that is often successful.


Anton Leykin, Institute for Mathematics and its Applications (IMA), Minneapolis, MN
Computing Embedded Solution Components via Deflation

This is a preliminary report on an algorithm that discovers embedded components in the solution set of a system of polynomial equations. A basic use for the method of deflation is to ``resolve''(*) the isolated multiple solutions of a polynomial system. Assuming the concept of a higher-order deflation we can construct such a ``resolution''(**) in a single step. Moreover, we may generalize this approach to treat positive-dimensional components occuring with multiplicity greater than one.

Now, let A be a component embedded into another component B of higher dimension. It can be shown that a deflation of a high enough order would lift these to components A* and B* not contained in each other.

(**) -- ``resolution'': an augmented system of equations in a higher-dimensional ambient space, such that the lifting of the original multiple solution is regular;
(*) -- ``resolve'': construct such a system.


Wenyuan Wu, University of Western Ontario, London, ON
Fast Prolongation Method for Partial Differential Equations

We present a fast symbolic-numeric method to compute Riquier Bases in implicit form for a class of PDE systems. They are dominated by pure derivatives in one of the independent variables and have the same number of PDE and unknowns. The method is successful provided the prolongations only with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are non-singular when evaluated at points on the zero sets defined by the functions of the PDE. For polynomially nonlinear PDE, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points as initial data for numerical solving of Differential Equations.