  We present a new, far simpler family of counter-examples to Kushnirenko's
Conjecture. Along the way, we illustrate a computer-assisted approach to
finding sparse polynomial systems with maximally many real roots, thus shedding
light on the nature of optimal upper bounds in real fewnomial theory. We use a
powerful recent formula for the A-discriminant, and give new bounds on the
topology of certain A-discriminant varieties. A consequence of the latter
result is a new upper bound on the number of topological types of certain real
algebraic sets defined by sparse polynomial equations, e.g., the number of
smooth topological types attainable in certain families of real algebraic
surfaces.
