On the Extended Hensel Construction and its Application to the Computation of Limit Points

Computing Puiseux series : a fast divide and conquer algorithm

“…Here we have ⌈F 2 ⌉ 7 = F 2 − X 8 and N (⌈F 2 ⌉ 7 ) = [(10, 0), (6, 1), (4, 2), (3,3), (2,5)]. But the edge [ (3,3), (2,5)] is not part of N 7 (F 2 ), as it belongs to 2 a + b = 9, and that there are points (i, j) so that 2 i + j ≤ 9 and j > 7: from the knowledge of ⌈F 2 ⌉ 7 , we cannot guarantee that N (F 2 ) contains an edge belonging to 2 a + b = 9. This is indeed wrong here, since N (F 2 ) = [(10, 0), (6, 1), (4, 2), (3,3), (0, 8)].…”

“…But the edge [ (3,3), (2,5)] is not part of N 7 (F 2 ), as it belongs to 2 a + b = 9, and that there are points (i, j) so that 2 i + j ≤ 9 and j > 7: from the knowledge of ⌈F 2 ⌉ 7 , we cannot guarantee that N (F 2 ) contains an edge belonging to 2 a + b = 9. This is indeed wrong here, since N (F 2 ) = [(10, 0), (6, 1), (4, 2), (3,3), (0, 8)]. Polynomial multiplication.…”

“…Proof. We follow the Las Vegas algorithm 3 given in [34,Section 2.2]. First, trace computation of the monomial basis takes O(M(d d Q )) operations in K (it is reduced to polynomial multiplication thanks to [27,Proposition 8]).…”

“…There are other methods to compute Puiseux series or analytic factorisation, as generalised Hensel constructions [3,20], or the Montes algorithm [4,28] (which works over general local fields). Several of these methods and a few others have been commented in previous papers by the first author [32,33].…”