Bivariate triangular decompositions in the presence of asymptotes

We show that the same worst-case complexities can be achieved even when the curves defined by the input polynomials may have common vertical asymptotes. We actually present refined complexities, O˜(dxdy3+dx2dy2) for the arithmetic complexity and O˜B(dx3dy3+(dx2dy3+dxdy4)τ) for the bit complexity, where dx and dy bound the degrees of P and Q in x and y, respectively. We also prove that the total bitsize of the decomposition is in O˜((dx2dy3+dxdy4)τ).