Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces
Abstract
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.
Keywords
Cite
@article{arxiv.math/0212178,
title = {Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces},
author = {Tien-Yien Li and J. Maurice Rojas and Xiaoshen Wang},
journal= {arXiv preprint arXiv:math/0212178},
year = {2007}
}
Comments
27 pages, 2 figures. Extensive revision of math.CO/0008069. To appear in Discrete and Computational Geometry. Technique from main theorem (Theorem 1) now pushed as far as it will go. In particular, Theorem 1 now covers certain fewnomial systems of type (n+1,...,n+1,m) and certain non-sparse fewnomial systems. Also, a new result on counting non-compact connected components of fewnomial hypersurfaces (Theorem 3) has been added