Sharp precision in Hensel lifting for bivariate polynomial factorization

Abstract

Abstract. Popularized by Zassenhaus in the seventies, several algorithms for factoring polynomials use a so-called lifting and recombination scheme. Concerning bivariate polynomials, we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored. Its cost is dominated by the computation of reduced echelon solution bases of linear systems. We show that our bound on precision is asymptotically optimal.

History of finite fields

“…But the first successes in the design and proofs of complete algorithms are due to van Hoeij [van Hoeij 2002] for the factorization in Z[x], and then to Belabas et al [Belabas et al 2009] for F (x) [y], with the logarithmic derivative recombination method, where the precision σ = deg f (deg f − 1) + 1 is shown to be sufficient in general. Then a precision linear in deg f in characteristic 0 or large enough characteristic has been shown to suffice in [Bostan et al 2004;Lecerf 2006].…”

Primary decomposition of zero-dimensional ideals over finite fields

“…The main problem is that, under a reduction at a prime ideal, a primary ideal may split into many primary ideals, so one needs to find the right combinations to get back to the original primary components. For polynomial factorization, which corresponds to the special case when I is generated by one polynomial, efficient algorithms were recently developed by van Hoeij [15] and Lecerf [16]. It is desirable to develop a similar theory for systems of multivariate polynomials.…”

Lifting and recombination techniques for absolute factorization

“…The deterministic, probabilistic and heuristic algorithms share the same main ideas that are adapted from the rational factorization algorithms of [48,49]. Our algorithms combine advantages of Gao's algorithm [26] and the classical Hensel lifting and recombination scheme.…”

“…Our algorithms combine advantages of Gao's algorithm [26] and the classical Hensel lifting and recombination scheme. This scheme was popularized by Zassenhaus [74,75] and is nowadays a cornerstone of the fastest rational factorization algorithms [38,3,48,49]. Now we sketch out the main stages of the algorithms.…”

“…Although this construction does not look intuitive at first sight, we will see that it is directly related to the recombination technique introduced in [48 ]. In addition, the notation is designed to be consistent with [48] but observe that *F *y = 1 and *F *x = − . For any i ∈ {1, .…”