Computing Puiseux series : a fast divide and conquer algorithm

A quasi-linear irreducibility test in K[[x]][y]

“…As δ = v i , we have min v i ≤ δ/d and the upper bound for η(F ) follows. If F is quasi-irreducible, then we have also v i ≤ η(F ) = N i by [21,Corollary 4]. As all v i 's are equal in that case, the lower bound follows too.…”

“…m 1 , z 1 are respectively α, 2 and 1. Applying results of [21,Section 3], one can show that an optimal truncation bound to compute G 1 is…”

On the complexity of computing integral bases of function fields

“…Then, we need to compute the Puiseux expansions η i of f at one root of each factor in S f ac , up to precision N = max i i =j v(η i − η j ). Using the algorithm of Poteaux and Weimann [21], the Puiseux expansions are computed up to precision N in O(n(δ + N )) field operations, where δ stands for the valuation of Disc(f ). Indeed, these expansions are computed throughout their factorization algorithm, which runs in O(n(δ + N )) field operations as stated in [21,Theorem 3].…”

“…Using the algorithm of Poteaux and Weimann [21], the Puiseux expansions are computed up to precision N in O(n(δ + N )) field operations, where δ stands for the valuation of Disc(f ). Indeed, these expansions are computed throughout their factorization algorithm, which runs in O(n(δ + N )) field operations as stated in [21,Theorem 3]. Therefore, in theory, we will see that computing the Puiseux expansions has a negligible cost compared to other parts of the algorithm since N ≤ n 2 .…”

“…Note that we count field operations and do not take into account the coefficient growth in case of infinite fields nor the field extensions incurred by the use of Puiseux series. We also made the choice not to delve into probabilistic aspects: all the algorithms presented here are "at worst" Las Vegas due to the use of Poteaux and Weimann's algorithm, see for instance [21,Remark 3].…”