Back to the roots: Solving polynomial systems with numerical linear algebra tools

Finding the roots of a set of multivariate polynomials has numerous applications in geometry and optimization, system and control theory, modeling and identification, statistics and bioinformatics, and many other scientific disciplines. It is an old yet fascinating problem, that has intrigued scientists throughout the ages, starting with the Greeks, over Fermat and Descartes, Newton, Leibniz, Bezout and many many others.

In this talk, we will elaborate on a research program, the objective of which is to translate the many – symbolic - algorithms from algebraic geometry, into numerical linear algebra algebra algorithms. Our talk develops ideas on three complementary levels:

• Geometric linear algebra, which deals with column and row vector spaces, dimensions, orthogonality, kernels, eigenvalue problems and the like
• Numerical linear algebra, where we conceptually deal with tools like Gram-Schmidt orthogonalization, the singular value decomposition, ranks, angles between subspaces, etc.
• Numerical algorithms, which implement the linear algebra tools into an efficient and numerically robust method. Here we can exploit matrix structures (e.g. Toeplitz or sparsity), investigate variations of iterative methods (e.g. power methods) or try to speeden up convergence (e.g. by FFT).