Gale dual systems and bounds in real algebraic geometry

Seminar on Applicable Algebraic geometry

2 August 2007
IMA PI Summer Graduate School on
    Applicable Algebraic Geometry.

    	Abstract:   Gale duality for polynomial systems is an elementary reformulation of
a system of polynomial equations as a system of equations involving rational
master functions in the complement of a hyperplane arrangement. Some properties of
the original system are easier to understand in the Gale dual system.

        In the first part of this talk, I will describe this Gale duality, look at some
examples of this construction, and give some elementary consequences. In the
second half, I will explain how to bound the number of real solutions to the Gale
dual system, and thereby obtain the current best bounds for the number of real
solutions to a system of polynomial equations. I will also explain how this gives
rise to a new numerical method for finding all real solutions to a system of
polynomial equations.