Complexity of OM factorizations of polynomials over local fields

Abstract

Let k be a locally compact complete field with respect to a discrete valuation v. Let O be the valuation ring, m the maximal ideal and F (x) ∈ O[x] a monic separable polynomial of degree n. Let δ = v(Disc(F )). The Montes algorithm computes an OM factorization of F . The singlefactor lifting algorithm derives from this data a factorization of F (mod m ν ), for a prescribed precision ν. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of O(n 2+ + n 1+ δ 2+ + n 2 ν 1+ ) word operations for the complexity of the computation of a factorization of F (mod m ν ), assuming that the residue field of k is small.

Computation of integral bases

“…By definition, the elements b κ,l ∈ B κ are given by b κ,l = g κ,l (θ) with g κ,l (x) ∈ A[x] monic of degree m < n P κ . In [1,Proposition 1.3] it is shown that all monic polynomials g ∈ A[x] of degree less than n P κ satisfy v P κ (g(θ))/e(P κ /p) ≤ μ for a certain constant μ which satisfies μ ≤ δ/n P κ . Hence, w P κ (b κ,l ) ≤ δ/n P κ , for all 0 ≤ l < n P κ .…”

“…The Montes algorithm has a cost of O n 2+ + n 1+ δ log q + n 1+ δ 2+ p-small operations [1,Thm. 5.15].…”

“…According to [1,Theorem 5.16], the cost of the computation of an Okutsu approximation Φ i with precision ν at P i (that is,…”

Fast Computation of the Roots of Polynomials Over the Ring of Power Series

“…Let us nally mention algorithms for polynomial factorization over local elds. Using the Montes algorithm [10], it is proved in [3] that one can compute a so-called OM-factorization of a degree n polynomial Q in F q x [ ] at precision d using O ∼ (n 2 ν + nν 2 + nν log(q)), where ν is the valuation of the discriminant of Q; the relation to basic root sets, de ned below, remains to be elucidated.…”

Genetics of polynomials over local fields

“…The concept of a type and the Montes algorithm were introduced in [13] for v a discrete valuation on a global field K. These results were reviewed in [5,6] and their computational implications were developed in a series of papers [2,7,8,9,14]. The derivation of these tools from the modern presentation of MacLane's valuations in the spirit of Vaquié, leads to a more elegant treatment of the subject and to its generalization to arbitrary discrete valued fields (K, v).…”