Fast Computation of the Roots of Polynomials Over the Ring of Power Series Vincent Neiger Johan Rosenkilde Eric´ Schost Technical University of Denmark Technical University of Denmark University of Waterloo Kgs. Lyngby, Denmark Kgs. Lyngby, Denmark Waterloo ON, Canada [email protected] [email protected] [email protected] ABSTRACT Problem 1. We give an algorithm for computing all roots of polynomials over a Input: univariate power series ring over an exact eld K. More precisely, • a precision d 2 Z>0, given a precision d, and a polynomial Q whose coecients are • a polynomial Q 2 K»»x¼¼»y¼ known at precision d. power series in x, the algorithm computes a representation of all Output: ¹ º ¹ ¹ ºº d power series f x such that Q f x = 0 mod x . e algorithm • (nite) list of pairs ¹fi ; ti º1≤i ≤` ⊂ K»x¼ × Z≥0 such that Ð ti works unconditionally, in particular also with multiple roots, where R¹Q;dº = 1≤i ≤`¹fi + x K»»x¼¼º Newton iteration fails. Our main motivation comes from coding theory where instances of this problem arise and multiple roots An algorithm solving this problem must involve nding roots must be handled. of polynomials in K»y¼. e existence, and complexity, of root- e cost bound for our algorithm matches the worst-case input nding algorithms for univariate polynomials over K depends on and output size d deg¹Qº, up to logarithmic factors. is improves the nature of K. In this paper, we assume that K is such that we upon previous algorithms which were quadratic in at least one of d can nd roots in K of a degree n polynomial in K»y¼ in time RK¹nº, and deg¹Qº. Our algorithm is a renement of a divide & conquer for some function RK : Z≥0 ! R; the underlying algorithm may algorithm by Alekhnovich (2005), where the cost of recursive steps be deterministic or randomized. For instance, if K = Fq , we can ∼ is beer controlled via the computation of a factor of Q which has take RK¹nº 2 O ¹nº using either a Las Vegas algorithm (in which a smaller degree while preserving the roots. case the runtime can be more precisely stated as O ∼¹n log¹qºº [19, Cor. 14.16]), or a deterministic one (with for instance a runtime p KEYWORDS O ∼¹nk2 pº, where we write q = pk , p prime [17]). Polynomial root-nding algorithm; power series; list decoding. We now state our main result: we separate the cost of the root- nding part of the algorithm, which may be randomized, and the 1 INTRODUCTION rest of the algorithm which is deterministic. In what follows, K is an exact eld, and K»»x¼¼»y¼ denotes the set of Theorem 1.2. ere is an algorithm which solves Problem 1 us- polynomials in y whose coecients are power series in x over K. ing O ∼¹dnº deterministic operations in K, together with an extra O¹dRK¹nºº operations, where n = deg¹Qº. Problem and main result. Given a polynomial in K»»x¼¼»y¼, we are interested in computing its power series roots to some precision, A cost in O ∼¹dnº is essentially optimal for Problem 1. Indeed, if as dened below. Q = ¹y − f1º · · · ¹y − fnº, for some power series f1;:::; fn such that fi − fj is a unit for all i , j, then the roots of Q to precision d are Definition 1.1. Let Q 2 K»»x¼¼»y¼ and d 2 Z> . A power series 0 all the power series of the form f + xd K»»x¼¼, for some i. In this f 2 K»»x¼¼ is called a root of Q to precision d if Q¹f º = 0 mod xd ; i case, solving Problem 1 involves computing all f mod xd , which the set of all such roots is denoted by R¹Q;dº. i amounts to dn elements in K. Our main problem (Problem 1) asks, given Q and d, to compute a Previous work. When the discriminant of Q 2 K»»x¼¼»y¼ has x- nite representation of R¹Q;dº; the fact that such a representation valuation zero, or equivalently, when all y-roots of Q are simple exists is explained below (eorem 2.8). In all the paper, we count jx=0 ∼ (as in the example above), our problem admits an obvious solution: operations in K at unit cost, and we use the so-O notation O ¹·º rst, compute all y-roots of Q in K, say y ;:::;y , for some to give asymptotic bounds with hidden polylogarithmic factors. jx=0 1 ` ` ≤ n, where n = degQ. en, apply Newton iteration to each of Permission to make digital or hard copies of all or part of this work for personal or these roots to li them to power series roots f1;:::; f` of precision classroom use is granted without fee provided that copies are not made or distributed / for prot or commercial advantage and that copies bear this notice and the full citation d; to go from precision say d 2 to d, Newton iteration replaces fi on the rst page. Copyrights for components of this work owned by others than the by author(s) must be honored. Abstracting with credit is permied. To copy otherwise, or Q¹fi º d republish, to post on servers or to redistribute to lists, requires prior specic permission fi − 0 mod x ; and/or a fee. Request permissions from [email protected]. Q ¹fi º ISSAC ’17, July 25–28, 2017, Kaiserslautern, Germany where Q 0 2 K»»x¼¼»y¼ is the formal derivative of Q. e boleneck © 2017 Copyright held by the owner/author(s). Publication rights licensed to ACM. 0 978-1-4503-5064-8/17/07...$15.00 of this approach is the evaluation of all Q¹fi º and Q ¹fi º. Using an DOI: hp://dx.doi.org/10.1145/3087604.3087642 algorithm for fast multi-point evaluation in the ring of univariate polynomials over K»»x¼]/(xd º, these evaluations can both be done Sudan’s and Guruswami-Sudan’s algorithms for the list-decoding in O ∼¹dnº operations in K. Taking all steps into account, we obtain of Reed-Solomon codes [8, 18] have inspired a large body of work, d ∼ the roots f1;:::; f` modulo x using O ¹dnº operations in K; this some of which is directly related to Problem 1. ese algorithms is essentially optimal, as we pointed out above. In this case, the operate in two stages: the rst stage nds a polynomial in K»x;y¼ total time for root-nding is RK¹nº. with some constraints; the second one nds its factors of the form us, the non-trivial cases of Problem 1 arise when Q jx=0 has y − f ¹xº, for f in K»x¼. multiple roots. In this case, leaving aside the cost of root-nding, e Newton-Puiseux algorithm can easily be adapted to compute which is handled in a non-uniform way in previous work, we are such factors; in this context, it becomes essentially what is known as not aware of an algorithm with a cost similar to ours. e best cost the Roth-Ruckenstein algorithm [16]; its cost is in O¹d2n2º, omiing bounds known to us are O ∼¹n2dº, obtained in [1] and with this cost the work for univariate root-nding. estimate being showed in [13], and O ∼¹nd2º, obtained in [4]. In the context of Sudan’s and Guruswami-Sudan’s algorithms, When Q jx=0 has multiple roots, a natural generalization of our we may actually be able to use Newton iteration directly, by ex- problem consists in computing Puiseux series solutions of Q. It is ploiting the fact that we are looking for polynomial roots. Instead then customary to consider a two-stage computation: rst, com- of computing power series solutions (that is, the Taylor expansions pute suciently many terms of the power series / Puiseux series of these polynomial roots at the origin), one can as well start from solutions in order to be able to separate the branches, then switch another expansion point x0 in K; if the discriminant of Q does not to another algorithm to compute many terms eciently. vanish at x0, Newton iteration applies. If K is nite, one cannot Most algorithms for the rst stage compute the so-called singular exclude the possibility that all x0 in K are roots of Q; if needed, one parts of rational Puiseux expansions [7] of the solutions. ey are may then look for x0 in an extension of K of small degree. Augot inspired by what we will call the Newton-Puiseux algorithm, that is, and Pecquet showed in [2] that in the cases appearing in Sudan’s Newton’s algorithmic proof that the eld of Puiseux series Khhxii is algorithm, there is always a suitable x0 in K. algebraically closed when K is algebraically closed of characteristic However, for example for the Wu list decoding algorithm [21] or zero [12, 20]. In the case of Puiseux series roots, one starts by for the list-decoding of certain algebraic geometry codes [13], one reading o the leading exponent γ of a possible solution on the does seek truncated power series roots. In this case, one may use Newton polygon of the input equation Q 2 Khhxii»y¼. e algorithm Alekhnovich’s algorithm [1, App.], which is a divide and conquer then considers Qˆ = Q¹xγ y)/xs 2 Khhxii»y¼, where s is the valuation variant of the Roth-Ruckenstein algorithm.