New Algorithms for Computer Algebra, Implementations and Applications

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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).