Extremal real algebraic geometry and A-discriminants

A. DICKENSTEIN, J. M. ROJAS, K. RUSEK , J. SHIH

Moscow Mathematical Journal

Independent University of Moscow

Lugar: Moscu; Año: 2007 vol. 7 p. 425 - 452

1609-4514

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 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.