Section:
New Results
Efficient algorithms for
rational first integrals
We presented in [29] fast algorithms for computing
rational first integrals with degree bounded by of a planar polynomial
vector field of degree . 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 and to a
deterministic algorithm for solving the problem in
arithmetic operations, where is the exponent of linear algebra. By
comparison, the best previous algorithm uses at least arithmetic operations. Our new algorithms are moreover very efficient in
practice.