# Applied Math Seminar: "Parametrized polynomial systems and real numerical algebraic geometry"

**Daniel Brake**

Postdoctoral Research Associate

Department of Applied and Computational Mathematics and Statistics

### "Parametrized polynomial systems and real numerical algebraic geometry"

Numerical Algebraic Geometry (NAG) is a set of theory and tools for studying systems of polynomials, and complements the symbolic methods. It is distinctly geometric in flavor, and is well-suited to a number of applications in engineering and science.

Systems of parametrized polynomial systems can be challenging to solve, both for symbolic methods and NAG, due to large number of symbolic objects, i.e. variables and parameters. The ubiquitous Monte Carlo method, or plain mesh-like discretization, can be a useful tool to study parametrized systems. Paramotopy, a program which solves systems over arbitrary samplings of a parameter space, has been used to study kinematic manipulators, chemical reaction networks, and biological multistability. Brake will talk about its theoretical framework, and applications of the software.

A complement to parametrized systems, under-determined systems have positive-dimensional components to their varieties. Over the complex numbers, components have generic behaviour, and are relatively easy to characterize, even if they have singularities. Over the real numbers, however, positive-dimensional varieties are difficult to study, most namely because the reals are a set of measure zero in the complexes, and the theoretical basis for NAG doesn’t hold. However, recent theoretical algorithmic advances have provided us with tools to study 1- and 2-dimensional real sets. In this vein, Brake will talk about Bertini Real, software which numerically `decomposes’ real curves and surfaces in any number of variables.

Originally published at acms.nd.edu.