Bivariate Triangular Decompositions in the Presence of Asymptotes

Abstract

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 bit complexity for computing such triangular decompositions when the curves defined by the input polynomials do not have common vertical asymptotes is in $\tilde{O}_B(d^6+d^5\tau)$ [Bouzidi et al. 2015, Prop. 16], where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. We show that the same worst-case bit complexity can be achieved even when the curves defined by the input polynomials may have common vertical asymptotes.We actually present a refined bit complexity in $\tilde{O}_B(d_x^3d_y^3 +(d_x^2d_y^3+d_xd_y^4)\tau)$ 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 $\tilde{O}((d_x^2d_y^3+d_xd_y^4)\tau)$