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

多元多项式方程组普遍存在于优化、统计学、生物学以及其他科学和工程领域。解决这样的系统是今天计算代数的基石，也是这个项目的主要焦点。该项目解决了多元多项式符号计算中的基本问题，特别关注出现在生物数据建模和数据挖掘中的非常大的系统。这样的系统是如此之大，以至于它们不能完全读入计算机的内存。该项目提出了使用概率和统计分析，巧妙地选择和采样较小的子系统，导致所需的解决方案。所探讨的问题是具有实际意义的基本问题。该项目还将在人力资源开发方面开展教育和培训活动。除了许多学生与研究人员合作外，该项目还包括一个暑期学校，从与这个跨学科研究项目相关的领域中学习基础数学概念。目标受众是研究生;学院将培养学生的社区意识，并加强进一步的跨学科合作。这个研究项目的方法解决系统的多项式方程的问题，并找到发电机多项式理想使用概率的方法-专注于提供低预期的运行时间和使用随机选择-计算代数问题，有很高的最坏情况的复杂性。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）来显著加速。由此产生的算法和软件将用于交换代数，统计，优化，图论和其他领域的大规模系统的多项式方程自然出现。该项目适应研究中的问题，已被用于计算几何和优化的采样技术。理论上预期的运行时间将与输入多项式的数量成线性关系。存在包括统计和优化的若干应用，其中关键应用方法依赖于算法来计算此类理想生成器。此外，该研究将增加组合结构在多项式计算问题中的使用，特别是用于计算Nullstelleneirs不可行性证书和syzygies。在图论中已经发现了应用，并且预期在组合学、编码理论和系统生物学中有进一步的应用。

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.