On the complexity of multivariate polynomial division by Joris van der Hoeven LIX, CNRS École polytechnique 91128 Palaiseau Cedex France Email:
[email protected] Web: http://lix.polytechnique.fr/~vdhoeven April 30, 2017 Abstract In this paper, we present a new algorithm for reducing a multivariate polynomial with respect to an autoreduced tuple of other polynomials. In a suitable sparse complexity model, it is shown that the execution time is essentially the same (up to a logarithmic factor) as the time needed to verify that the result is correct. Keywords: sparse reduction, complexity, division, algorithm A.M.S. subject classi cation: 68W30, 12Y05, 68W40, 13P10
1 Introduction Sparse interpolation [1, 5, 2, 13] provides an interesting paradigm for efficient computations with multivariate polynomials. In particular, under suitable hypothesis, multiplication of sparse polynomials can be carried out in quasilinear time, in terms of the expected output size. More recently, other multiplication algorithms have also been investigated, which outperform naive and sparse interpolation under special circumstances [14, 12]. An interesting question is how to exploit such asymptotically faster multiplication algorithms for the purpose of polynomial elimination. In this paper, we will focus on the reduction of a multivariate polynomial with respect to an autoreduced set of other polynomials and show that fast multiplication algorithms can indeed be exploited in this context in an asymptotically quasi-optimal way. Consider the polynomial ring K[x] = K[x1; :::; xn] over P an effective eld K with an eective zero test. Given a polynomial P = i2Nn Pi xi = P i1 in n / 0g the supi1;:::;in 2N Pi1;:::;in x1 xn , we call supp P = fi 2 N : Pi = port of P . The naive multiplication of two sparse polynomials P ; Q 2 K[x] requires a priori O(jsupp P j jsupp Qj) operations in K. This upper bound is sharp if P and Q are very sparse, but pessimistic if P and Q are dense. . This work has been supported by the ANR-09-JCJC-0098-01 MaGiX project, the Digiteo 2009-36HD grant and Région Ile-de-France.
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Section 1
Assuming that K has characteristic zero, a better algorithm was proposed in [2] (see also [1, 5] for some background). The complexity of this algorithm can be expressed in the expected size s = jsupp P + supp Qj of the output (when no cancellations occur). It is shown that P and Q can be multiplied using only O(M(s) log s) operations in K, where M(s) = O(s log s log log s) stands for the complexity of multiplying two univariate polynomials in K[z] of degrees < cB x ei if k 2 Fin(flBi ; :::; lBb g) and i k (x ) = i is minimal with lBi 4 k > :e k x otherwise b+1 By linearity, this operator extends to K[x] X (P ) = Pi (xi): i2supp P
In particular, (cA x ) yields the leading term of the extended reduction ^ the corresponding operator from K[x z ¡] (Q1; :::; Qb; R). We also denote by ¡ b+1 to K[x z ] which sends P^ to (P ). Now let Bi = Bi ¡ cBi xlBi for each i. Then lA
(Qi Bi)k = (Qi Bi)k + (Qi)k ¡lBi cBi for each i 2 f1; :::; bg and k 2 Nn. The equation (Q1 B1 + + Qb Bb + R)k = Ak can thus be rewritten as (Q1)k¡lB1 cB1 + ::: + (Qi)k¡lBb cBb = (A ¡ Q1 B1 ¡ ¡ Qb Bb)k Using the operator this equation can be rewritten in a more compact form as (Q1; :::; Qb ; R) = (A ¡ Q1 B1 ¡ ¡ Qb Bb): The tagged counterpart ¡ ¡ ^ 1; :::; Q ^ b; R ^ = ^ A ^¡Q ^1 B ^1 ¡ ¡ Q ^b B ^b Q
is recursive, whence the extended reduction can be computed using b multi^1 B ^1; :::; Q ^b B ^b. With A; Bi ; Qi and R as in the variate relaxed multiplications Q previous section, theorem 4 therefore implies: Theorem 7. We may compute the extended reduction of A with respect to B in time O(SM(jB1 + Q1j) log (B1 + Q1) + + SM(jBb + Qb j) log (Bb + Qb) + jRj):
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Remark 8. Following remark 2, we also notice that A, the Qi and R may be replaced by vectors of polynomials in K[x]m (regarded as polynomials with coe cients in Km), in the case that several polynomials need to be reduced simultaneously.
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