New Algorithms for Computer Algebra, Implementations and Applications

我的研究领域是计算机代数：粗略地说，这是一些具有代数色彩的数学问题的研究，可以通过精确计算的方式来处理。这个主题实际上涵盖了一系列问题，从算法开发到复杂性分析，再到低级代码优化技术。*我在这个提案中的广泛目标是为计算机代数的核心问题(如求解方程系统)开发高效的算法解决方案家族，并实现高性能。我将在这个提案中解决的确切问题是，针对具有额外结构(例如，在优化中表现出来)的重要系统系列的更好算法，用于各种形式的运动规划的算法，以及最后受到纠错理论启发的解决问题的算法。这些都是计算机代数中的基本问题；它们的解决方案将代表着该领域本身的重大进步，并有许多应用(如上所述，在运动规划或纠错方面)。

My field of research is computer algebra: roughly speaking, this is the study of some mathematical problems with an algebraic flavor, that can be dealt with by means of exact computations. This subject actually covers a spectrum of questions, from algorithm development to complexity analysis, to low-level code optimization techniques.***My broad objective in this proposal is the development of efficient families of algorithmic solutions, with high-performance implementations, for questions at the heart of computer algebra, such as solving systems of equations. The precise questions I will address in this proposal are better algorithms for important families of systems that carry extra structure (which show up for instance in optimization), algorithms for various forms of motion planning and finally algorithms to solve problems inspired by error-correction theory. These are all fundamental questions in computer algebra; their solutions would represent significant progress for the field itself, and also have many applications (as mentioned above, in motion planning or error-correction).**