算法少单项式理论的第一步
摘要
少单项式理论始于仅依变量个数与单项式项数对多项式方程组实根个数的显式界。此处我们迈出合乎逻辑的下一步,研究相应的存在性问题:令 FEAS_R 表示判定给定具有整数系数的多元多项式方程组是否有实根的问题。我们描述了当限制于由恰含 m 个单项式项的单个 n 元多项式构成的输入时,m 大到使 FEAS_R 为 NP 难的相变:对 m<=n+2(任意固定 n)为多项式时间,而对 m<=n+n^{epsilon}(n 变化且任意固定 epsilon>0)为 NP 难。由于 FEAS_R 与 A-判别式之间的重要联系,我们进而研究了一些可在多项式时间内判定符号的新型 A-判别式族。(A-判别式包含所有已知结式作为特例,而后者是算法代数几何中的核心对象。)来自丢番图逼近的 Baker 定理成为关键工具。在此过程中,我们还导出了 n 元 (n+2)-单项式实零集的新定量界。
引用
@article{arxiv.math/0411107,
title = {First Steps in Algorithmic Fewnomial Theory},
author = {Frederic Bihan and J. Maurice Rojas and Casey E. Stella},
journal= {arXiv preprint arXiv:math/0411107},
year = {2007}
}
备注
25 pages, 5 figures. MAJOR revision of earlier version: (1) Frederic Bihan is a new co-author, (2) Theorem 1 is strengthened with a much sharper complexity threshold, (3) bounds on connected components from Theorem 2 are dramatically sharpened, (4) Theorem 3 strengthened considerably, (5) Corollary 1 removed, but theorem of Karpinski+Shparlinski on univariate discriminants is inserted, to clarify complexity comparisons