Collaborative Research: Randomized and Structure-Based Algorithms in Commutative Algebra

多元多项式方程组在最优化、统计学、生物学等科学和工程领域中普遍存在。求解这类系统是当今计算代数的基石，也是这个项目的主要焦点。该项目解决了多变量多项式符号计算中的基本问题，尤其对出现在生物数据建模和数据挖掘中的超大型系统特别感兴趣。这样的系统太大了，以至于不能完全读入计算机的内存。该项目建议使用概率和统计分析来巧妙地选择和抽样较小的子系统，以实现所需的解决方案。所攻击的问题是具有实际意义的基本问题。该项目还将开展人力资源开发方面的教育和培训活动。除了许多学生与研究人员一起工作外，该项目还包括一个暑期班，介绍与这一跨学科研究项目相关领域的基本数学概念。目标受众是研究生；学院将在学生中培养社区意识，并进一步加强跨学科合作。这项研究项目探讨了求解多项式方程系统的问题，以及使用概率方法为具有高最坏情况复杂性的计算代数问题寻找多项式理想的生成器的问题--专注于提供较低的预期运行时间和随机选择的使用。该项目使用某些问题族的基本组合结构(例如，多项式系统可行性)，以实现显著的加速。由此产生的算法和软件将用于交换代数、统计学、最优化、图论和其他自然产生大规模多项式方程系统的领域。该项目适应了正在研究的问题，这是一种已用于计算几何和优化的抽样技术。理论上期望的运行时间将与输入多项式的数量成线性关系。有几种应用，包括统计和优化，其中的关键应用方法依赖于算法来计算这样的理想生成器。此外，这项研究将增加组合结构在多项式计算问题中的使用，特别是在计算Nullstellensatz不可行性证书和合子时。已经在图论中找到了应用，并期望在组合学、编码理论和系统生物学中有更多的应用。

Systems of multivariate polynomial equations are ubiquitous in optimization, statistics, biology, and other fields of science and engineering. Solving such systems is a cornerstone of computational algebra today and the main focus of this project. The project addresses fundamental problems in symbolic computation with multivariate polynomials, with particular interest in very large systems that appear, for example, in biological data modeling and data mining. Such systems are so large that they cannot be completely read into the computer's memory. This project proposes the use of probabilistic and statistical analysis to cleverly select and sample smaller subsystems that lead to the desired solution. The problems attacked are fundamental questions of practical relevance. The project will also have educational and training activities in the development of human resources. In addition to many students working with the investigators, the project includes a Summer School on the foundational mathematical concepts from the areas relevant to this interdisciplinary research project. The target audience is graduate students; the School will foster a sense of community among the students and enhance further interdisciplinary collaboration. This research project approaches the problem of solving systems of polynomial equations, and of finding generators for polynomial ideals using a probabilistic method -- focusing on providing low expected runtime and the use of random choices -- for computational algebra problems that have high worst case complexity. The project uses the underlying combinatorial structure of certain families of problems (e.g. polynomial system feasibility) for significant speed-up. The resulting algorithms and software will be of use in commutative algebra, statistics, optimization, graph theory, and other fields where large-scale systems of polynomial equations arise naturally. The project adapts to the problem under study a sampling technique that has been used in computational geometry and optimization. The theoretically expected running time will be linear in the number of input polynomials. There are several applications including statistics and optimization, where key applied methods rely on algorithms to compute such ideal generators. Furthermore, the research will increase the use of combinatorial structures in polynomial computational problems, in particular, for the calculation of Nullstellensatz infeasibility certificates and syzygies. Applications have been found in graph theory, and further applications are expected in combinatorics, coding theory, and systems biology.