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Speaker: J. Maurice Rojas, Texas A&M University

Title: Extremal Real Algebraic Geometry and A-Discriminants


Descartes' Rule tells us that the maximal number of real roots of a sparse polynomial f (in one variable) depends solely on the number of monomial terms of f. It took over three centuries to extend this result to arbitrary systems of multivariate equations, and even now there is no optimal upper bound for two variables. In this talk, we present a new, far simpler family of counter-examples to a conjectural upper bound of Kushnirenko. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. A consequence of our techniques is a new upper bound on the number of topological types of real algebraic sets defined by sparse polynomial equations, e.g., the number of attainable cardinalities for certain real zero sets. We assume no background in algebraic geometry. We also point out that this work arose from a summer REU project, and is joint with Alicia Dickenstein, Korben Rusek, and Justin Shih.

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