On the complexity of the Lickteig–Roy subresultant algorithm

Abstract

In their 1996 article, Lickteig and Roy introduced a fast divide and conquer variant of the subresultant algorithm which avoids coecient growth in defective cases. The present article concerns the complexity analysis of their algorithm over eective rings endowed with the partially dened division routine. This leads to new convenient complexity bounds for gcds, especially when coecients are in abstract polynomial rings where evaluation/interpolation schemes are not supposed to be available.

On the Complexity Exponent of Polynomial System Solving

“…In the latter proof we could have appealed to faster algorithms to build irreducible polynomials, such as the ones of [16,60,67]. In addition, faster algorithms are known to obtain in specific cases; see [9,28,51]. For simplicity we have preferred to restrict to general well known algorithms, because the complexity bound of Lemma 3.4 is to be used only when l is rather small.…”

“…First we need to compute Res t (ȏ (t), q (t)) along with the corresponding Bézout relation that yields p ≔ ȏ −1 mod q . We wish to apply a fast subresultant algorithm, namely either the one of [21, chapter 11] or the one of [51]. For this purpose, we need to ensure the following properties:…”

Accelerated tower arithmetic

“…Given an element a ∈ 1 , the characteristic polynomial of a is the characteristic polynomial of the multiplication endomorphism by a in 1 . If = is a field, then this polynomial can be computed in softly quadratic time by means of a resultant (see for instance [44,Corollary 29]). Shoup has also designed a practical randomized algorithm to compute minimal polynomials for the case when has sufficiently many elements, with an expected complexity of O M(d) d + d 2 .…”

“…In a similar way as for the computation of gcds in section 2.5, it is possible to avoid divisions during the computation of resultants and subresultants. However, the required adaptations of the algorithms from [44] are a bit more technical than the mere replacement of Euclidean divisions by pseudo-divisions. For this reason, we have not further optimized the number of divisions in our complexity bounds.…”

Composition Modulo Powers of Polynomials

“…Proof. The rst two bounds directly rely on the formula (x) = Res z (g(z) ¡ x; h(z)) by using [20,Corollary 29]. The third bound is obtained by computing Res z (g(z) ¡ a; h(z)) for n + 1 values of a in A, from which is interpolated using O(M A (n) log n) additional operations in A.…”