DMCA
Factorization of Univariate Polynomials with Rational Coefficients
BibTeX
@MISC{Novocin_factorizationof,
author = {Andrew Novocin and Mark Van Hoeij},
title = {Factorization of Univariate Polynomials with Rational Coefficients},
year = {}
}
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Abstract
Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples where this latter a is ten times smaller than the a used in Zassenhaus’ algorithm. Just multiply a small irreducible polynomial by a big one. (Of course a needs to be large enough not only to find g2,...,g k, but also large enough to prove that g1,...,g k are irreducible. More precisely, a needs to be large enough to solve the combinatorial problem. However, using [van Hoeij] this can usually be done with a much smaller a than what is used in Zassenhaus ’ algorithm.) An easy way (called Early Termination) to prevent lifting too far is to do these two steps after each Hensel lift: 1. Try to solve the combinatorial problem using [van Hoeij]
Keyphrases
univariate polynomial rational coefficient combinatorial problem zassenhaus algorithm van hoeij p-adic accuracy large factor hensel lifting true factor cpu time big one potential factor current implementation hensel lift early termination ten time small irreducible polynomial hensel lift current implementation p-adic integer small factor local factor modular image easy way factor g1
