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。

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).