Extremal real algebraic geometry and $\mathcal A$-discriminants

We present a new, far simpler family of counterexamples 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 $\mathcal A$-discriminant, and give new bounds on the topology of certain $\mathcal A$-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.