Following the list of topics scheduled for today, we should start considering regular systems of empirical multivariate polynomials. Before we leave the beautiful fantasy world of pure mathematics entirely, we continue the topic of last Friday, of lecture 26 on homotopies and polyhedral methods.

In essence -- giving a sketch of a constructive proof of Koushnirenko's theorem -- is not that far from the scheduled topic: with "regular" systems we assume our systems are square and we look only after isolated zeros. Moreover, since homotopies rely on the continuity of zeros, they are very well suited to deal with "empirical" polynomials.

A very nice introduction to the ideology of fewnomials is the paper by Bernd Sturmfels on "Polynomial Equations and Convex Polytopes" which appeared in the American Mathematical Monthly, volume 105, number 10, pages 907-922, December 1998.