Bivariate triangular decompositions in the presence of asymptotes
Résumé
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
$\widetilde{O}((d_x^2d_y^3+d_xd_y^4)\tau)$.
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