Extremal Real Algebraic Geometry and A-Discriminants

Details Extremal Real Algebraic Geometry and A-Discriminants Extremal Real Algebraic Geometry and A-Discriminants

2006-09-18

arxiv.org/abs/math/0609485v2

Mathematics

Algebraic Geometry

24 pages, 13 figures, final version with several improvements and small corrections

Scientific paper

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.

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