三项式曲线交与 m 项超曲面的实连通分支计数
代数几何
2007-05-23 v1 组合数学
摘要
我们证明任意一对二元三项式在正向象限中至多具有 5 个孤立根。此前与多项式次数无关的最佳上界要大得多，例如根据 Khovanski 的一个著名一般结果为 248832（仅针对非退化根）。我们的上界是紧的，允许实指数，允许退化，并可推广到某些 n 元少项方程组，给出的改进相对于早期上界在单项式个数上是指数级的。我们还推导了单个 n 元 m 项式的实零点集连通分支数的类似锐化上界。
引用
@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}
}
备注
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