Overall Objectives
Research Program
Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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Section: New Results

Efficient algorithms for rational first integrals

We presented in  [29] fast algorithms for computing rational first integrals with degree bounded by N of a planar polynomial vector field of degree dN. The main novelty is that such rational first integrals are obtained by computing via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity Õ(N2ω) and to a deterministic algorithm for solving the problem in Õ(d2N2ω+1) arithmetic operations, where ω is the exponent of linear algebra. By comparison, the best previous algorithm uses at least dω+1N4ω+4 arithmetic operations. Our new algorithms are moreover very efficient in practice.