10:00am - 12:00pmNumerical methods in algebraic geometry
Chair(s): Jose Israel Rodriguez (UW Madison, United States of America), Paul Breiding (MPI MiS)
This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods.
(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)
Certification of approximate roots of exact ill-posed polynomial systems
Agnes Szanto
NCSU
In this talk, I will survey some of our recent work on certifying approximate roots of exact polynomial systems and will describe some applications. In particular, we will concentrate on systems with root multiplicity, and show ways to certify approximations to singular roots, as well as their multiplicity structure. The difficulty lies in the fact that having singular roots is not a continuous property, so traditional numerical certification techniques do not work. Our certification methods are based on hybrid symbolic-numeric techniques. This is joint work with Jonathan Hauenstein and Bernard Mourrain.
Numerical Implicitization
Justin Chen
Georgia Tech
It is increasingly important nowadays to perform explicit computations on varieties, even in the realm where symbolic (e.g. Grobner basis) methods are too slow. We give an overview of the Macaulay2 package NumericalImplicitization, which aims to provide numerical information about images of varieties, such as dimension, degree, and Hilbert function. We also discuss some changes and additions to the package, such as improvements to point sampling, completions of partial pseudo-witness sets, and parallelization. This is joint work with Joe Kileel.
The Distribution of Numbers of Operating Points of Power Networks
Julia Lindberg
Wisconsin Institute for Discovery
The operating points of an n-node power network are real solutions of the power flow equations, a system of 2n-2 quadratic polynomials in 2n-2 variables. Our work finds the distribution of the number of real solutions, which is important in determining the stability of the network. In general, the number of nontrivial operating points equals the number of real solutions of a single polynomial. We use this polynomial to visualize regions with a fixed number of solutions, finding that some cluster around hyperplanes.