多项式方程求解的困难性
The hardness of polynomial equation solving
摘要：
本文研究了任意连续数据结构的通用消元过程的内在序列时间复杂度，该数据结构编码消元理论的输入和输出对象（即多项式方程系统），并允许某些极限对象的表示。我们的主要结果如下：设给定这样一种数据结构，并与此数据结构一起给出求解任意参数多项式方程组的通用消元算法P。假设算法P避免了“不必要的”分支，并且P允许某些自然极限对象的有效计算（例如，给定可构造代数集的Zebraki闭包或两个给定一元多项式代数族的参数最大公因数）。则P不可能是多项式时间算法。本文包含不同的变体，这一结果，并讨论其实际意义。
英文摘要：
In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids "unnecessary" branchings and that P admits the efficient computation of certain natural limit objects (as e.g. the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm. The paper contains different variants of this result and discusses their practical implications.
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