Trace Tests in Numerical Algebraic Geometry

Numerical algebraic geometry uses numerical algorithms to study algebraic varieties, which are sets defined by polynomial equations.

It is becoming a core tool in applications of algebraic geometry outside of mathematics. Its fundamental concept is a witness set which gives a representation of a variety that may be manipulated on a computer. A trace test is used to verify that a witness set is complete and provide stopping criteria for algorithms. Recently, there have been further developments in computational algebraic geometry to study varieties in a product space, which is the natural setting for solutions of parameterized polynomial systems. In this talk, we discuss some of these developments and a new trace test for these varieties. We demonstrate the methods' effectiveness in examples from various areas of science and statistics.