  We generalize the classical lifting and recombination scheme for rational and
absolute factorization of bivariate polynomials to the case of a critical
fiber. We explore different strategies for recombinations of the analytic
factors, depending on the complexity of the ramification. We show that working
along a critical fiber leads in some cases to a good theoretical complexity,
due to the smaller number of analytic factors to recombine. We pay a particular
attention to the case of polynomials that are non degenerate with respect to
their P-adic Newton polytopes.
