Speaker:
J. Maurice Rojas, Texas A&M University
Title:
Extremal Real Algebraic Geometry and ADiscriminants
Abstract:
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 counterexamples
to a conjectural upper bound of Kushnirenko. Along the way, we illustrate
a computerassisted 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.
