Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

Details Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

2002-12-12

arxiv.org/abs/math/0212178v1

Mathematics

Algebraic Geometry

27 pages, 2 figures. Extensive revision of math.CO/0008069. To appear in Discrete and Computational Geometry. Technique from m

Scientific paper

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.

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