Bivariate triangular decompositions in the presence of asymptotes

Given two coprime polynomials $P$ and $Q$ in $\mathbb{Z}[x,y]$ of degree at most $d$ and coefficients of bitsize at most $\tau$, we address the problem of computing a triangular decomposition $\{(U_i(x),V_i(x,y))\}_{i\in\cal I}$ of the system $\{P,Q\}$. The state-of-the-art worst-case complexities for computing such triangular decompositions when the curves defined by the input polynomials do not have common vertical asymptotes are $\widetilde{O}(d^4)$ for the arithmetic complexity and $\widetilde{O}_B(d^{6} +d^{5}\tau)$ for the bit complexity, where $\widetilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. 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, $\widetilde{O}(d_xd_y^3+d_x^2d_y^2)$ for the arithmetic complexity and $\widetilde{O}_B(d_x^3d_y^3 + (d_x^2d_y^3+d_xd_y^4)\tau)$ for the bit complexity, where $d_x$ and $d_y$ bound the degrees of $P$ and $Q$ in $x$ and $y$, respectively. We also prove that the total bitsize of the decomposition is in $\widetilde{O}((d_x^2d_y^3+d_xd_y^4)\tau)$.