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Fast Computation of Generic Bivariate Resultants∗

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Fast Computation of Generic Bivariate Resultants∗ Fast computation of generic bivariate resultants∗ JORIS VAN DER HOEVENa, GRÉGOIRE LECERFb CNRS, École polytechnique, Institut Polytechnique de Paris Laboratoire d'informatique de l'École polytechnique (LIX, UMR 7161) 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France a. Email: [email protected] b. Email: [email protected] Preliminary version of May 21, 2020 We prove that the resultant of two “sufficiently generic” bivariate polynomials over a finite field can be computed in quasi-linear expected time, using a randomized algo- rithm of Las Vegas type. A similar complexity bound is proved for the computation of the lexicographical Gröbner basis for the ideal generated by the two polynomials. KEYWORDS: complexity, algorithm, computer algebra, resultant, elimination, multi- point evaluation 1. INTRODUCTION The efficient computation of resultants is a fundamental problem in elimination theory and for the algebraic resolution of systems of polynomial equations. Given an effective field 핂, it is well known [10, chapter 11] that the resultant of two univariate polynomials P,Q∈핂[x] of respective degrees d⩾e can be computed using O˜ (d) field operations in 핂. Here the soft-Oh notation O˜ (E) is an abbreviation for E(log E)O(1), for any expression E. Given two bivariate polynomials P,Q∈핂[x,y] of respective total degrees d⩾e, their 2 resultant Resy(P, Q) in y can be computed in time O˜ (d e); e.g. see [20, Theorem 25] and references in that paper. If d=e, then this corresponds to a complexity exponent of 3/2 in terms of input/output size. An important open question in algebraic complexity theory is whether this exponent can be lowered. In the present paper, we consider the case when P and Q are “sufficiently generic”. If the coefficients of P and Q are chosen at random in a finite field 핂= 픽q with sufficiently many elements, then this will be the case with high probability. Under a suitable hypoth- esis of “grevlex-lex-generic position” (defined below) and assuming the random access memory (RAM) bit complexity model, our main result is the following theorem: THEOREM 1. Let 휖 > 0 be a fixed rational number. Let P, Q ∈ 핂[x, y] be two polynomials of respective total degrees d ⩾ e over a finite field 핂 = 픽q. If P and Q are in grevlex-lex-generic position, then Resy(P,Q) can be computed in expected time O((delog q)1+휖)+O˜ (d2 log q), using a randomized algorithm of Las Vegas type. ∗. This paper is part of a project that has received funding from the French “Agence de l'Innovation de Défense”. 1 2 FAST COMPUTATION OF GENERIC BIVARIATE RESULTANTS Figure 1. Illustration of the Gröbner stairs for P and Q in generic position with respect to <grevlex (left) and <lex (right) in the case when d =5 and e =3. A major result in a similar direction has recently been obtained by Villard [25]. For a general effective field 핂, and under different genericity assumptions, he proposed an algorithm that computes the resultant in y of two polynomials P,Q∈핂[x,y] of degree dx 2−1/휔 1+o(1) in x and degree dy in y using (dx dy ) operations in 핂. Here 휔 is the usual expo- nent for matrix multiplication (such that two n × n matrices over 핂 can be multiplied using O(n휔) operations in 핂). Le Gall has shown in [19] that one may take 휔<2.373. Relationship to Gröbner bases Resultants are related to elimination theory. Throughout the paper, we assume that the reader is familiar with the basic theory of Gröbner bases, as found in standard text books [4, 10], so we only briefly recall basic terminology. Let 핂 still be a general effective field. Two common monomial orderings on the polynomial ring 핂[x, y] are the lexicographical ordering <lex and the reverse graded lexicographical ordering <grevlex defined by i j k l x y <lex x y ⟺ j<l∨(j=l∧i<k) i j k l x y <grevlex x y ⟺ (i+ j<k+l∨(i+ j=k+l∧ j<l)). We say that P and Q are in lex-generic (resp. grevlex-generic) position if the leading mono- mials of the reduced Gröbner basis of I with respect to <lex (resp. <grevlex) coincide with the ones that we would obtain when taking symbolic parameters for the coefficients of P and Q; see Figure 1. We say that P and Q are in grevlex-lex-generic position when they are both in lex-generic and grevlex-generic position. Notice that we do not require the ideal I ≔(P,Q) to be radical over the algebraic closure of 핂. The relationship between resultants and Gröbner bases is the following: if P and Q are in lex-generic position, then the reduced Gröbner basis of I with respect to <lex con- sists of the minimal polynomial of x, which is a constant multiple of Resy(P,Q), and the polynomial y− U(x) for some U ∈핂[x] with deg U <de; see section 4.1. Assuming that P and Q are in grevlex-generic position, the recent result from [12] is an algorithm to compute a concise representation [12, Definition 14] of the Gröbner basis 2 of I with respect to <grevlex using O˜ (d ) operations in 핂, while conserving its main prop- erties; see section 3. Note that the Gröbner basis in its usual representation generically requires Θ(d2 e) storage, so its computation is too expensive for our purposes. If we were able to rapidly convert a Gröbner basis for <grevlex into a new one for <lex, then this would allow us to compute resultants in softly linear time. Unfortunately, tradi- tional “change-of-ordering” algorithms such as the FGLM algorithm [6, 7] rely on linear algebra, and do not run in softly linear time. For our proof of Theorem 1 in section 6, we will instead rely on a bivariate counterpart of Kedlaya–Umans' algorithm for mod- ular composition [16]. This technique does not work for general effective fields 핂, which explains the restriction to the case when 핂=픽q is a finite field in Theorem 1. JORIS VAN DER HOEVEN, GRÉGOIRE LECERF 3 Specific fields For 핂 = ℚ, Mehrabi and Schost [21] showed how to compute a basis of I ≔(P,Q) with respect to <lex by means of a probabilistic algorithm of Monte Carlo type with a nearly optimal bit complexity bound. Their bound is optimal when the coeffi- cients grow as expected in the worst case and their method depends on Kedlaya–Umans as well. This approach has been generalized to higher dimensions in [14]. The proba- bilistic aspect comes from the need of a sufficiently generic linear change of the variables. The best known deterministic complexity bound can be found in [2]. Over finite fields, Poteaux and Schost [22, Theorem 1.2] achieved the computation of bivariate lexicographical bases with bit complexity (d e)1+휖 O˜ (log q) in the special case when P or Q belongs to 픽q[y], provided that the underlying characteristic is greater than d e, and that I is radical, yet without genericity assumption. Their method extends to an arbitrary number of variables. Outline of our contribution The proof of Theorem 1 relies on a sequence of reductions, using a novel combination of classical and more recent techniques. In sections 2 and 3, we first recall basic notations and the required results from [12] about concise Gröbner bases. Assuming from there on that P and Q are in grevlex-lex-generic position, we recall in section 4 how to reduce the computation of the resultant to the computation of the min- imal polynomial of the multiplication endomorphism by x+I in 픸≔핂[x,y]/I. This min- imal polynomial can be computed with high probability using Wiedemann's algorithm, provided that we have an algorithm for the transposed map of evaluating a univariate polynomial at x + I in 픸. This kind of strategy has been used several times before in computer algebra [15, 22, 23, 24]; see also [10, chapter 12, section 4]. The evaluation of a univariate polynomial at x+I in 픸 can be regarded as a bivariate modular composition problem. Exploiting the fact that multiplication in 픸 is fast (thanks to the concise Gröbner basis representation), we show in section 5 how to reduce this problem to multivariate multipoint evaluation. For this reduction, we mostly follow Kedlaya and Umans, along the same lines as in the proof of [16, Theorem 3.1] for uni- variate modular composition; refinements can be found in [13]. At that point, we restrict ourselves to the case when 핂 is a finite field, so as to benefit from Kedlaya and Umans' fast algorithm for multipoint evaluation and its transpose [16]. In section 6, this allows us to conclude the proof of Theorem 1. The final section 7 contains a few further notes and directions for future research. In particular, we show that the full lexicographical Gröbner basis can be computed in a similar time as the resultant, with ideas similar to [22] and [8, Algorithm 2]. Finally we quantify the genericity hypotheses in a more precise manner. 2. PRELIMINARIES Computational model Throughout this paper, 핂 is an effective field. Most of our algo- rithms work in the algebraic complexity models of straight-line programs (SLPs) or com- putation trees [3], in which execution times correspond to the required number of field operations in 핂.
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