数学少项理论的第一步
First Steps in Algorithmic Fewnomial Theory
摘要：
Fewnomial理论开始明确的界限-仅在数量的变量和单项条款-对一些实根系统的多项式方程。在这里，我们采取下一个逻辑步骤调查相应的存在性问题：让FEAS_R表示的问题，决定是否一个给定的系统的多元多项式方程的整数系数有一个实根或没有。当m大到足以使FEAS_R成为NP-难时，我们描述了一个相变，当限制输入由一个具有正好m个单项项的n变量多项式组成时：m<=n+2（对于任何固定的n）的多项式时间和m<=n+n^{n}（对于n变化和任何固定的n>0）的NP-难。由于FEAS_R与A-判别式之间的重要联系，我们研究了一类新的A-判别式，它们的符号可以在多项式时间内确定。（A-判别式包含所有已知的结式作为特例，后者是算法代数几何的核心。从丢番图逼近中得到的贝克定理是一个重要的工具。在此过程中，我们还得到了新的定量界限的实零集的n-变量（n+2）-诺姆。
英文摘要：
Fewnomial theory began with explicit bounds -- solely in terms of the number of variables and monomial terms -- on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEAS_R denote the problem of deciding whether a given system of multivariate polynomial equations with integer coefficients has a real root or not. We describe a phase-transition for when m is large enough to make FEAS_R be NP-hard, when restricted to inputs consisting of a single n-variate polynomial with exactly m monomial terms: polynomial-time for m<=n+2 (for any fixed n) and NP-hardness for m<=n+n^{epsilon} (for n varying and any fixed epsilon>0). Because of important connections between FEAS_R and A-discriminants, we then study some new families of A-discriminants whose signs can be decided within polynomial-time. (A-discriminants contain all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) Baker's Theorem from diophantine approximation arises as a key tool. Along the way, we also derive new quantitative bounds on the real zero sets of n-variate (n+2)-nomials.
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