Bivariate triangular decompositions in the presence of asymptotes

Abstract

Given two coprime polynomials P and Q in Z[x,y] of degree at most d and coefficients of bitsize at most τ, we address the problem of computing a triangular decomposition {(Ui(x),Vi(x,y))}i∈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 O˜(d4) for the arithmetic complexity and O˜B(d6+d5τ) for the bit complexity, where O˜ refers to the complexity where polylogarithmic factors are omitted and OB 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, 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).