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.
In this paper we present an algorithm that computes the genus of a global function field. Let F/k be function field over a field k, and let k 0 be the full constant field of F/k. By using lattices over subrings of F , we can express the genus g of F in terms of [k 0 : k] and the indices of certain orders of the finite and infinite maximal orders of F . If k is a finite field, the Montes algorithm computes the latter indices as a by-product. This leads us to a fast computation of the genus of global function fields. Our algorithm does not require the computation of any basis, neither of finite nor infinite maximal order.
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