Counting Isolated Roots of Trinomial Systems in the Plane and Beyond
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 counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, 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 new bounds on the number of real connected components of fewnomial hypersurfaces.
Cite
@article{arxiv.math/0008069,
title = {Counting Isolated Roots of Trinomial Systems in the Plane and Beyond},
author = {Tien-Yien Li and J. Maurice Rojas and Xiaoshen Wang},
journal= {arXiv preprint arXiv:math/0008069},
year = {2007}
}
Comments
18 pages, submitted for publication. Further streamlining and clarifications made. There is also a new figure illustrating the optimality of a variant of Rolles' Theorem we use