Things Your Mother Should Have Told You About Infinity

J. Maurice Rojas (MIT)

Note: This lecture will be occuring in a special day (Friday), time (1:00 pm), and room (2-338)!

Toric variety methods have recently proven to be extremely useful for dealing with computational problems involving polynomial systems. However, the resulting methods are, at least initially, geared towards affine space minus the coordinate hyperplanes.

We explain a few tricks which allow us to correctly extend toric methods to affine space. In particular, we obtain

• a generalization of the recent stable mixed volume bounds for the number of isolated affine roots of a polynomial system,
• a combinatorial condition for when our root counting formula is exact, and
• a new resultant operator tailored for computations in affine space.

Along the way, we obtain a concrete explanation for how roots ``go to infinity'' in certain homotopy-based algorithms for solving polynomial systems. This explanation offers an alternative algebraic approach to the recent polyhedral homotopy methods.