On the Complexity of the Montes Ideal Factorization Algorithm

Abstract

Let p be a rational prime and let Φ(X) be a monic irreducible polynomial in Z[X], with nΦ = degΦ and δΦ = v _p (discΦ). In [13] Montes describes an algorithm for the decomposition of the ideal $p\mathcal{O}K$ in the algebraic number field K generated by a root of Φ. A simplified version of the Montes algorithm, merely testing Φ(X) for irreducibility over Q _p , is given in [19], together with a full Maple implementation and a demonstration that in the worst case, when Φ(X) is irreducible over Q _p , the expected number of bit operations for termination is O(nΦ^3 + ε δΦ^2 + ε). We now give a refined analysis that yields an improved estimate of O(nΦ^3 + ε δΦ + nΦ^2 + ε δΦ^2 + ε) bit operations. Since the worst case of the simplified algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the complete Montes algorithm.