2006/09/18 by Alicia Dickenstein, Dickenstein, Alicia, J. Maurice Rojas +5
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Polynomial and algebraic computation #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.math/0609485
openalex publication_date 2006/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.