四环上的Algebraic Geometry和可处理性的前沿
Algebraic Geometry Over Four Rings and the Frontier to Tractability
摘要：
我们提出了一些新的和最近的算法结果多项式系统解决各种环。特别是，我们提出了一些最好的最近的界限：（a）计算复杂的代数集的维数的复杂性，（b）高度的零维部分的代数集在C上，和（c）的连接组件的半代数集的数量。我们还提出了一些结果，显着降低的复杂性，决定在C和Q上的超曲面交的空虚，广义黎曼假设的真理。此外，我们状态的可判定性的前缀\exists \forall \exists和\exists \exists \forall \exists，量化的正整数的一些最新进展。作为应用，我们给出了三元Hilbert第十问题与代数曲线上整点高度界的一个结果。这篇论文是根据在会议上提出的三个讲座相应的这卷会议录。讲座的标题是“一些加速计算代数几何”，“丢番图问题几乎在多项式层次，”和“曲线，曲面，和前沿的不可判定性。''
英文摘要：
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic set, (b) the height of the zero-dimensional part of an algebraic set over C, and (c) the number of connected components of a semi-algebraic set. We also present some results which significantly lower the complexity of deciding the emptiness of hypersurface intersections over C and Q, given the truth of the Generalized Riemann Hypothesis. Furthermore, we state some recent progress on the decidability of the prefixes \exists\forall\exists and \exists\exists\forall\exists, quantified over the positive integers. As an application, we conclude with a result connecting Hilbert's Tenth Problem in three variables and height bounds for integral points on algebraic curves. This paper is based on three lectures presented at the conference corresponding to this proceedings volume. The titles of the lectures were ``Some Speed-Ups in Computational Algebraic Geometry,'' ``Diophantine Problems Nearly in the Polynomial Hierarchy,'' and ``Curves, Surfaces, and the Frontier to Undecidability.''
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
