New Algorithms for Computer Algebra, Implementations and Applications

我的研究领域是计算机代数：粗略地说，这是研究一些带有代数味道的数学问题，可以通过精确计算来处理。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。***我在这个提议中的广泛目标是开发高效的算法解决方案族，具有高性能的实现，用于解决计算机代数的核心问题，例如求解方程组。我将在本提案中解决的确切问题是，为携带额外结构的重要系统家族提供更好的算法（例如在优化中出现），为各种形式的运动规划提供算法，最后是解决受纠错理论启发的问题的算法。这些都是计算机代数中的基本问题；他们的解决方案代表了该领域本身的重大进步，并且也有许多应用（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).**