Параллельное вычисление резольвент Лагранжа с помощью мультирезольвент Текст научной статьи по специальности «Математика»

UDC 519.688

PARALLEL COMPUTATION OF LAGRANGE RESOLVENTS BY MULTI-RESOLVENTS © Philippe Aubry, Annick Valibouze

LIP6, UPMC, 4, place Jussieu, F-75252 Paris Cedex 05, France, e-mail: Philippe.Aubry@upmc.fr, Annick.Valibouze@upmc.fr

Key words: Lagrange resolvent; Galois group; galoisian ideal; triangular ideal; double class; parallel computation.

The goal of this paper is the parallel computation of Lagrange resolvents of a univariate polynomial. The computation of Lagrange resolvents of a univariate polynomial has significance for Galois Theory. Since Lagrange’s algorithms, many other algorithms for computing some particular resolvents, called absolute, were developed from the fundamental theorem of symmetric functions. The algebraic algorithms for non absolute resolvents are few and recent because they use galoisian ideals that were introduced recently. However these algorithms become time and space consuming when the degree of the polynomial increases. This motivates their parallelization. Rennert proposed a multi-modular method for computing absolute resolvents of polynomials with integer coefficients. We show that the same techniques can be extended to any resolvent. This method is naturally parallelizable. Moreover, we give a decomposition formula of resolvents which makes possible another level of parallelization. This leads to an algorithm with a doubly parallel character.

1 Introduction

The Lagrange resolvent of a univariate polynomial f is a fundamental tool in Galois theory (see [1] and [2]). It is a univariate polynomial obtained from a multivariate polynomial f

group stabilizing the multivariate polynomial 0 used for the transformation; hence, they

f

0

The parallel method that we describe is inspired by Rennert’s work (see [4]) for the restricted case of the resolvents relative to the symmetric group, called absolute resolvents. Nevertheless Rennert’s method cannot be adapted automatically to the general case of resolvents relative to subgroups of the symmetric group. The reader can refer to Example 1 that illustrates one of the simplifications existing when the reference group is the symmetric group. This paper does not only extend to any resolvent the multimodular parallélisation proposed by Rennert, but presents another level of parallélisation thanks to a new decomposition formula of resolvents given in Theorem 1. Moreover, the theoretical study that leads to this decomposition, together with the description of a strategy for the parallel computation, bring to the subject greater clarity.

Section 2 introduces galoisian ideals and Lagrange resolvents with some properties. Section 3 establishes Theorem 1 in which the resolvent splits into factors corresponding to double

f

f

These latter, called multi-resolvents, may be computed in parallel. Section 5 applies the method of Section 4 to the important case of irreducible (or reducible) polynomials over the rational

f

coefficients are integers. Since the image of f modulo p is reducible for many prime integers p, we can perform the computation of a Lagrange resolvent of f in Fp [x] by the above parallel computation with multi-resolvents. The Lagrange resolvent of f in Z[x] is finally lifted from those in Fp[x] by using the Chinese Remainder Theorem, This multimodular method is clearly “doubly“ parallel since the Lagrange resolvents of f in Fp[x] are computed independently. Finally, Section 6 is devoted to the parallel algorithm description.

Throughout this paper, k is a perfect field, k an algebraic closure of k, fa, square-free univariate polynomial of k[x] with degree n and a = (a\,... ,an) in k is a tuple of the n f

General notation For a variety V C ~k\ the ideal Id(V) of V is the set of polynomials with coefficients in k vanishing on each element of V .Let I be an ideal of k[x1,... ,xn], the algebraic variety V(I) of I is the set of point in kn where every polynomial in I vanishes. The symmetric group of degree n is denoted by Sn . Given two ideals I and J, the injector Inj (I, J) of I in J is the set of elements of Sn sending each element of I in J. The subset StabSn (I) := Inj (I, I) of Sn is a group, called the stabilizer of I in Sn (in literature, it is also called the decomposition group of I ). For H < Sn and a G Sn , Ha = a Ha-1.

2 The Lagrange resolvent

The results not referenced or proved can be found in [5] where galoisian ideals are introduced. The maximal ideal of a-relations M = Id(a) has as stabilizer

G = StabSn (M) ,

which is the Galois group of a in k.

By the natural k-morphism x^ ^ ai from k[x-\_,..., xn] to k[a1,..., an] = k(a1,..., an),

the field k(a1,..., an) of the roots of ^ to the quotient ring k[x1,..., xn]/M,

M

G

ascending chain

Il C I2 C ■ ■ ■ C M

I1

possible to take the ideal Id(Sn.a), called the ideal of symmetric relations, which is generated by the Cauchy moduli, a triangular Groebner basis obtained by divided differences from the polynomial f. The resolvents have a double interest: construct a generator of Ii+1 tom Ii and simultaneously exclude some groups to be the Galois group of a by using the matrices of

groups. More generally, the resolvents are intensively used in numeric and algebraic methods for computing the Galois group alone.

Let us define galoisian ideals and their injectors. Let L be a set of permutations of Sn such that L = GL (i.e. G < L when L is a group). The ideal I of the variety L.a is called a galoisian ideal, L is its injector in the galoisian ideal M ; the algebraic variety of I is V(I) = L.a . Note that G is the injector of M in itself and M = Id(G.a) = Id(a).

When the injector L of I in M is a group, the galoisian ideal I is said pure. A galoisian

ideal is pure if and only if L equals the stabilizer of I in Sn ; it is itself equivalent to the

inclusion of the Galois group G in this stabilizer. When I is pure V(I) = L.p for each P E V(I), It is proved in [6] that a pure galoisian ideal is generated by a separable triangular set of polynomials; such an ideal is said triangular.

Definition 1 The L -relative resolvent of a by 0 E k[xl, . . . ,xn] is the polynomial

R©,L,a = n (x - y(a)) .

^GL.©

When I is pure, the resolvent does not depend on the choice of a in V(I) ; it thus can be denoted by R©,I.

The characteristic polynomial of the multiplicative endomorphism ©induced by 0 in k[xi,... ,xn]/I is a power of the resolvent :

— pcard(H) rn

X© ,i = R©,i (l)

where H < L is the stabilizer of 0 in L ( 0 is called an L -relative H -invariant). By linear

algebra x©1 belongs to k[x]. As the field k is perfect, R©,I lies also in k[x]; moreover, if it

is square-free then R©, I is the minimal polvnomial of © , the square-free form of x© I ■

In next section, we will apply Sentence 2 below to a subgroup K of L in order to compute L-relative resolvents. For this reason, we prefer to use respectively K and J = Id(K.a) instead of L and I = Id(L.a) in the rest of the present section.

Let K < Sn and t E &n. Then we have

Id(KT 1 .(T.a)) = t-l .Id(K.a) . (2)

Indeed, for each t E &n:

Id(T-1 Kt.(t.o_)) = Id(KT.a) = t-l .Id(K.a) .

Sentence 1 Let K < Sn, J = Id(K.a) and t E Sn . The galoisian ideal t-l.J is pure with stabilizer KT if and only if G < K, where G is the Galois group of a .

Proof 1 The Galois group of T.a is the conjugate GT 1 of G, and the condition GT 1 < KT 1 is equivalent to G < K . From Identity (2), the group KT and T.a define the galoisian ideal t-l.J.

11 KT is a group, the galoisian ideal t-l.J is pure with stabilizer KT if and only if the

Galois group of T.a is a subgroup of KT (see [5]); this is equivalent to G < K .

Sentence 2 Let K < Sn, J = Id(K.a) and t E Sn . Assume that the Galois group G of a K

R©,T-1.j = RT.©,J .

Proof 2 The characteristic polynomial of © in k[xl,...,xn ]/t -l.J is

X©,t-1.j = n (x - a.0(P))

a€KT-1

for any P E V(t-l.J) since, by Lemma 1, the galoisian ideal t-l.J is pure with, stabilizer KT 1 . Thus, for any P E V(t-l.J)

R©,t-1.j = ^ (x - v.0(P)) (3)

aeKT-1 /StabRT-i (&)

Moreover,

StabKT-i (0) = [а Є т lKr | a.O = 0}

= т l[p Є K | т 1 рт.0 = 0}т

= т-1[р Є K | р.(т.0) = т.0}т

= StabK (т.0)т (4)

We can choose в = т.а Є V(т-1.J) = Kт 1 .(т.а) (see Identity (2)). With the notations

р = тат-1 and S = StabK(т.0), Identities (3) and (4) imply

Re,T-1.J = Д (x - т-1рт.0(т.а))

peK/S

= П (x — тт-1 рт.0(а))

peK/S

= П (x — р.(т.0)(а)) (5)

peK/S

= RT.&,J

Remark 1 From Identities (3) and (5), the following equality can be deduced more generaly for any subgroup K of Sn :

^ (x - a.0(p))= ^ (x - p.(T.0)(a)) (6)

aGKT-1 /St-1 PGK/S

for any P E V(t-l.J) where t E &n and S = StabK(t.0) .

3 Double classes and resolvents

We are interested in computing the resolvent R@j, where I is a pure galoisian ideal of stabilizer

L. We show how a resolvent can be factored relatively to a double transversal. This can lead

to decomposing the resolvent into a product of resolvents with smaller degrees, in particular when the polynomial f is reducible.

Following the notations of previous section, H < L is the stabilize r of 0 in L .Let K be another subgroup of L. The relation RK)H = R defined in L by

a R t if aH fl Kt = 0

is an equivalence relation. The class of a is called a double class of L modulo K and H and satisfies the following proposition:

Sentence 3 Let a,T E L . Then a R t if and only if t E K aH .

Let us assume that we know a double transversal

K\L/H = {ti,... ,Tm}

of L modulo K and H, that is a set of representants of the equivalence classes of R. We thus have

m m

L.0 = y KTiH.0 = y KTi.0 .

i=1 i=1

In order to decompose each term of the above union, we introduce the subgroups

Hi := K f HTi

of K for i E [l,m],

Lemma 1 Let Ti E L and Hi as above. If 0 is an L -relative H -invariant then Ti.0 is a K Hi

Proof 3 For each permutation a of Hi, there exists a1 in H such, that

aTi.0 = TialT~1Ti.0 = Ti.0 . (7)

Therefore Ti.0 is invariant under the action of Hi,.

Now let a E K which leaves Ti.0 invariant. We show that a belongs to Hi,. From aTi.0 = Ti.0 we deduce that 0 is invariant under the a ction of T-1aTi. Then T~1aTi E H, in other words a E tíHt~1 .

From Identity (7), we find by decomposing K according to a left transversal K/H¿ of K :

KTi.0 = (K/Hi)HiTi.0

= (K/Hi)Ti.0 . (8)

Lemma 1 ensures us that the set (K/Hí)tí.0 has the same cardinality as the set K/H

a

and a' of K such that aTi,.0 = a' n.0 then a-1a' is in ^because it leaves n.0 invariant.

Then a and a' belong to the same left classe of K modulo K, Finally, we can write

Sentence 4 Let K\L/H = {r\,..., Tm} and H = K if T'Ht- for i G [1, m\ ; we have

m

LB = |J(K/Hi)Ti.e , (9)

i=1

where the union is disjoint.

Proof 4 Equality (9) follows from identities (1) and (8). Moreover, assume that there exists two permutations a and o' of K such that ot'B = a'TjB. Then aTi G afTjH and consequently Ti G KTjH, that is contradictory with definition of the double transversal.

Sentence 5 Let H = StabL(B), K\L/H = {t1, ... ,Tm} , Ki = KTi , Hi = K if HTi and

Hi = K,i if H for i G [1, m\ . Then

m

= H n (x - aTiB(a)) (10)

i=1 ae(K/Hi) m

= n n (x - aB(Ti.a)) (11)

i=1 ve(Ki/H)

with, Hi = StabK(n.Q) and H' = StabKi(B) for i G [1,m].

Proof 5 From Proposition 4, we express the resolvent as follows:

m

R&,1 = n n (x - ^(o))

i=1 *£(K/Hi)Ti.e

m

= H n (x - aTiB(a))

i=1 ae{K/Hi)

by Lemma 1. By the same lemma Hi = StabK(t'B) and one can easily verifies that Hi = StabKi (B). Then Equality (11) follows from Remark 1.

The resolvent Re,i is algebraically computable by the algorithms in [6] or [7] based on successive resultants when a triangular basis of I is given, Anvwav their costs may be dramatically reduced if it is possible to split the computation in several resolvents relative to galoisian ideals with generators of smaller degrees. By the independance of these factors the computation becomes parallelisable. Following these considerations an effective decomposition of Rej is given below.

Theorem 1 Let H = Stabl(B) and K\L/H = {^,..., Tm} . If G < K and if a triangular basis of J = Id(K.a) is given, then for each, i G [1,m] the resolvent ReT-ij is computable

i

mm

Re,i = H Re,T-1.J = n RTi.e,J . (12)

i=1 i=1

Proof 6 Let Ji = T- 1 .J. If G < K then by Lemma 1, the ideals J and Ji are pure with respective stabilizers K and KTi . Therefore Sentence 2 and Relations (10) and (11) leads to Identities (12). Furthermore R@T — J is computable since it is expressed as a resolvent relative to J.

4 Case of reducible polynomials and application to Fp[x]

As the goal of this paper is to compute the resolvent Re,I by multimodular techniques when the base field is Q , we will apply Theorem 1 to Fp [x] for / reducible over Fp , Consequently, in this section the polynomial / is supposed to be reducible over k , In order to split the computation of the resolvent Re,I we intend to determine a subgroup K of L containing the Galois group G, and such that the triangular basis of the associated galoisian ideal J = Id(K.a) is quickly computable.

Let / = /1 • • • /r, /i G k[x], For each i in [1,r], we denote by di the degree of / and by Gj, the Galois group (over k) of ai, a di -tuple of the di roots of /,

It is well known that there exists a conjugate GT of the Galois group G, t G &n, such that GT < G1,...,r = G1 x • • • x Gr . For the the goal of the paper, it is sufficient to consider the case where G1,...,r < LT , For a = T-1 , the group G1.. , r satisfies the following condition:

G < Gl. ,r < L. (13)

We first show how a triangular basis of the ideal I' = Id(G1. ,r.a) can be obtained. Let M1,..., Mr be th e r maximal galoisian ideals of the respective ^-relations. For each i G [1, r], we can rename the variables appearing in the triangular basis of M as a tuple ^ , and consider the ideal M' in the ring k[yi]. In this context, let us denote by Ti(yi) a triangular generating set of M' ,

Let T be the triangular set formed by t he union of T]_,... ,Tr , and T' obtained by replacing in T each variable yi,j by a variable xs such that this substitution is one-to-one (among the set of variables y^- and the set of variables x^) and such that the pure galoisian ideal I' generated by T' has G1.. , r as stabilizer,

T' T''

T

y1,1 * x1i y1,2 * x2, ... , Vr,dr : xn ;

T'' G1,...,r

Gi

give us:

T' = a~1.T" .

As G < G1.., r any group K such that

Gl.,,,, r <K <L (14)

is a candidate for our goals, and the galoisian correspondance about ideals implies:

I = Id(L.a) c J = Id(K.a)

C I' = Id(Gi..,r .a)

C M = Id(G.a) .

Example 1 When K = Sdl x ... x Sdr the triangular basis of the ideal J is the union of the triangular bases of the galoisian ideals of symmetric relations of the polynomials f1,..., fr , given respectively by the Cauchy moduli of fi (see [8]). If moreover L is the symmetric group then we are in the particular situation studied by N. Rennert in [4] in order to compute absolute resolvents ; in this case, T' = T'' (i.e. a = id ) and the condition (13) is satisfied for a = id.

To simplify the rest of this presentation, we assume without lost of generality that

G < Gi..,r < L

and that the n-tuple a of roots of f in V (I ) is ordered as well : the di-tupl e ai of roots of fi stands after the roots of fi_1 and before the roots of fi+1.

Let U1,... ,Ur be r groups such that Gj, < Ui < Sdi for i = 1,... ,r and U1 x.. .xUr < L .

We can set

K = U1 x ... x Ur .

The union of the triangular Groebner bases of the ideals Id(Ui.ai) forms a triangular basis of J

I'

Practically, we choose groups Ui as small as possible such that the computation of Re,K.a '1S the fastest ineludind the cost of a triangular basis of Id(Ui.ai).

Application to Fp[x]

The coefficients of f belongs to Fp , where p is a prime integer. In this particular case, the respective Galois groups Gi of fi are the cyclic groups Cdi of degree di. Denote by Mi the (maximal) galoisian ideal of ^-relations- The varietv of M^s Cdi.a^. As the Galois group is cyclic, the triangular basis of Mi can be computed easily tom the irreducible factors of fi in Fp[x]/ < fi >. Note that it is not necessary to factorise f completely (see [3]), The best choice is U-i = Cdi for i = 1,..., r . We have just to find a E Sn such that

K =(Cdi x ■ ■ ■ x Cdr Y < L. (15)

5 Computation by multimodular technique

Let f E Z[x] be any polynomial of degree n with n distinct roots in C , We want to compute R = R@,L,a for a group L containing the Galois group G of a .

Suppose that we computed the resolvent R modulo prime numbers p1,... ,ps such that the product p1 ■ ■ ■ ps is greater than the double of the maximal absolute value of the coefficients of RR

In this section, we have to compute efficiently R modulo дате prime p and to establish a bound on the coefficients of R. In addition, we give a certification to stop the algorithm before the bound is reached.

Assume that the integer p does not divide the discriminant of f. Such an integer, called unramified, exists since f is square-free, Set g = g mod p for any polynomial g. Recall the essential following theorems:

Theorem 2 (Dedekind, [9] ) Let f (x) G Z [x] be a polynomial of degree n with n distinct roots in C and le t G be the Galois group of f ove r Q in Sn (i.e. for a ny a). If p is unramified and f = f l ■ ■ ■ f r with fi irreducible о ver Fp of degree di, then there exists т G G with a cycle decomposition o\.. .ar with ai of length di.

The tuple (dl,..., dr) of Theorem 2 is called the cycle pattern of a and the decomposition type

f

Theorem 3 (Frobenius Density Theorem, [10]) Let (dl,..., dr) be a partition of n . Then, the relative density of the set of primes p for which f modulo p has a given decomposition type (d\,... ,dr ) exists and equals 1/\G\ times the number of a G G with cycle pattern (d\,... ,dr ).

Note that Frobenius Density Theorem is extended by Tchebotarev Density Theorem [11].

5.1 Computing R modulo p

In Fp[x], the polynomial f factorises into r irreducible factors as follows:

f = fl ■■■ fr

where deg(fi) = di. When r = l the prime integer p is “bad“’ and we throw this integer. Frobenius Density Theorem 3 shows the density of “good“ primes.

Let Gi = Cdi be the Galois group over Fp[x] of a di-tuple â of roots of fi, i G [l,r], and G be the Galois group of â ove r Fp[x] ; â can be chosen such that G < Gl x ■ ■ ■ x Gr . As p is unramified, for some a G &n, by Dedekind Theorem 2, this inclusion follows:

(G)a < (Gl x ■ ■ ■ x Gr)a < G < L .

We are exactly in the situation in which the computation by decomposition of the L -relative resolvent of â can be performed efficiently (see Section 4).

5.2 Bounding the coefficients of the resolvent R@;1

For the general case of relative resolvents, we just have to modify the Rennert’s formulae ([4]) by replacing the symmetric group Sn , stabilizing the ideal of symmetric relations, by the group L, stabilizing the galoisian ideal I. This leads to the following expression for a bound on the f

B(R) = (Ce (Cf + l))dD6

where Cg is the largest coefficient of a p olvnomial g in Z[xb ... ,xn], Dg its total degree, and d = deg(R) = [L : H] .

5.3 Efficient probabilistic solution with certification

Since the above bound may need approximatively [L * H]De prime integers to obtain the resolvent, a probabilistic approach is interesting to limit these necessary primes. Let us denote by

Rq = R mod q .

When q = p1 • • • pj , where p1... ,pj are prime numbers, the polvnomial Rq can be lifted by the Chinese Remainder Algorithm from the polynomials Rpi, A classical wav to obtain the resolvent with high probability consists in returning Rq as soon as Rq = Rq' where q' = p1... ,pj ,Pj+1.

We actually use another test to stop the computation. In [12], the condition R(B) G I is exploited as a certification for numerical computations of resolvents. Following this idea, even though q is smaller than 2B(R), we cut the computation when Rq = 0 modulo I. As the ideal I is triangular, this test is reduced to only n euclidean divisions in Q[x^ ..., xn].

6 The parallel algorithm

p f Kp

L chosen in order to construct the resolvent Rp by means of the double transversal Kp\L/H . The cardinality of this double transversal will be denoted by mp .

A total parallel computation of the multi-resolvent Rp would require mp + 1 processors :

mp

of Theorem 1, For a given p, following Theorem 1, each branch computes a resolvent RTi.&,J ■ In practice, these computations of resolvents take always similar times. It seems impossible to characterize a different behaviour since the timings depend essentially on the respective stabilisators of Ti.0, which are pairwise conjugate groups. For instance, the method of [6] computes the resolvent by successive resultants of the polynomial (x — Ti.0) with respect to

J

to significant different timings as we compute a resolvent by Ti.0 or bv Tj.0. However, we cannot assure rigorously that the degree of parallelism is mp + 1.

The difficulty of this algorithm is closely related with the two levels of parallélisation of the method. When the lifting of the resolvent by Chinese remainder is based on s prime integers p1,...,ps to obtain a certified result, more than mpi + ... + mps processors are required for a total parallélisation. In practice, this generally leads to a partial parallélisation, and it is not easy to decide and handle the repartition of the processors with respect to the two different tasks. Since a solution may be lifted with high probability without the computation of every Rpi,..., Rps , we naturally privilege the total computation of some resolvents Rp , We suppose that S + 1 processors proc(0),... ,proe(S) are available for the computation. The execution is controled by the master processor proc(0).

Step 1 /* This step refers to Section 5 */

Processor proc(0)

1.1 Compute a list P of unramified prime integers p1,... ,ps such that p1 ■ ■ ■ ps > 2B(R).

1.2 S0 := min{s,S}.

Step 2 /* This step refers to Section 4 */

For each processor proe(i), i=l to S0 , do

2.1 Compute fPi

2.2 Factorise /pi in FPi [x] into irreducible factors r

G1, . . . , Gr

2.4 Compute Kpi (see Condition (15))

2.5 Compute a double transversal D of Kpi\L/H and mi *= mpi

2.6 For j = 1,..., r compute the maximal ideals Mj (actually their triangular basis)

J Kpi Mj

2.8 Send pi,KPi, D, J,mi to the principal processor proe(0)

Step 3

/* This step refers to Section 5 for proc(S) and to Section 3 for the others,

R

Note 2: We estimate that the modular resolvents will be computed in similar times, */ Processor proc(0)

> Receive the above respective data from proe(l) to proe( S0) and stores them in a list t

> While (t is not empty) do

- Compute the largest integer u G [1, S0] and the number S1 of processors such that

S1 = (m1 + 1) + ... + (mu + 1) < S

- Delegate the computation of the resolvents RPi (i G [1,u]), to the u processors

proe( N.i) (1 ^ i ^ u), where N1 = 1 and N = m1 + • • • + mi-1 + i if i > 1

- Receive a boolean from proc(S) in the variable STOP

- If STOP=true Then Send a Signal to proe( i) (1 ^ i ^ S — 1) ; Break ; End If

ut

- End While

> If STOP=true Then Receive the resolvent R from proc(S) ; Return R ; End If

> Delete p1,... ,pSo in the list P / * see Step 1 */

> s *= length (P)

> S0 *= min{s,S — 1}

> Return to Step 2

Processors proe(j), 1 ^ j ^ S1

f* When the S1 processors needed to obtain the modular resolvents receive the Signal from proc(0) their computations are simultaneously stopped, */

> If j G {N1,...,Nu} Then

- Distribute the computation on the mi processors proe( Ni + 1), ,.., proc( Ni+1 — 1) by

sending to them p-i, 0, KPi, H, J and t G D , double transversa 1 with #D = mi,

- Wait to gather the results

- Compute the product of these results

- Send the product to proc(S)

>

- Receive p-i, 0, KPi, H, J, t from some proe( N)

- Perform the computation of Re,T.J mod pi mentioned in Theorem 1

- Send R@,t.j mod p' to proc( N)

>

Processor proc(S)

/* Note 1: this is actually repeated until the boolean variable STOP is set to true, meaning that

R

may be performed independently. Remark that proc(S) could be replaced by a set of processors

R

Note 2: we introduce variables m and F that contain respectively the product of the primes already taken into account and the current value of Rm, */

>

>

>

- Receive p1,... ,pu and RP1,..., RPu from proe(0)

- Let m *= mp1 ■ ■ ■ pu

- Compute Rm tom F and the RPi by Chinese Remainder Theorem

- F *= Rm

- If m > 2B(R) or F(0) G I Then

STOP := True End If

- Send STOP to proc(0)

- End While >F

Conclusion and further developments

A part of this paper has been employed to establish Theorem 1 and to solve the problems of its application. These problems did not appear in Rennert’s paper for computing absolute resolvents (see Exemple 1), It is important to note that our method is also more efficient in his context (L = Sn). Indeed, the group K in the decomposition

m

R&,I = RTi.@,J

i=1

of the theorem where J = Id(K.a), may be much smaller than the product of symmetric

K

groups in the computation of the modular resolvents. Furthermore, in Rennert’s paper the parallel strategy of his algorithm is not described though his practical exemple is sufficient to illustrate the interest of the methodology.

The modular computations have a double interest implying the double parallel character of the method. Indeed, each modular resolvent is computed in parallel and Theorem 1 is applied to compute in parallel factors of each modular resolvent. This doubly parallel character makes the implementation rather technical. However it opens a field of investigations and development of efficient strategies on how to optimize the distribution of the work on the processors between the different parallélisations involved in Steps 2 and 3 of the algorithm.

The modular computation of a resolvent produces some useful informations :

• One can apply the standard technique to exclude some groups to be the Galois group because each factorisation of f mod p (p an unramified prime) gives a subgroup of the Galois group of f (see Theorem 2),

• A partial factorisation of the resolvent Rpi on F^x] is a by-product of the algorithm; these factorisations of modular resolvents could be memorized in view of a future factorisation of the resolvent R on Z[x] that is useful for computing minimal polynomials of algebraic numbers, the Galois group or galoisian ideals.

In some recent works, the double classes of groups have been exploited to study galoisian ideals and resolvents. The theoritical results of the present paper show their importance. This tool should probably leads to some future developments and understanding in Galois theory.

References

1, Lagrange J. Réflexions sur la résolution algébrique des équations, Prussian Academy, 1770,

2, Galois E. Oeuvres Mathématiques, éditées par la SMF, Paris: Gauthier-Villars, 1897,

3, Valibouze A. Sur les relations entre les racines d’un polynôme//Acta Arithmetica, 2008, V, 131.1. P. 1-27.

4, Rennert N. A parallel multi-modular algorithm for computing Lagrange resolvents//J. Symb, Comput. 2004. V. 37. N. 5. P. 547-556.

5, Valibouze A. Etude des relations algébriques entre les racines d’un polynôme d’une variable Bull. Belg, Math, Soc, Simon Stevin, 1999, V, 6, N, 4, P, 507-535,

6, Aubry P., Valibouze A. Using Galois ideals for computing relative resolvents//J. Symbolic Comput. 2000. V. 30. N. 6. P. 635-651.

7, Aubry P., Valibouze A. Calcul algébrique efficace de résolvantes relatives. Archives HAL-CNRS, 2009. URL: http://hal.archives-ouvertes.fr/hal-00406357/en/.

8. Rennert N., Valibouze A. Calcul de résolvantes avec les modules de Cauchv//Experiment, Math. 1999. V. 8. N. 4. P. 351-366.

9. Dedekind R. Sur la théorie des nombres entiers algébriques. Paris:Gauthier-Villars, 1877.

10. Frobenius F. G. Uber beziehungen zwischen den primidealen eines algebraischen körpers und den Substitutionen seiner gruppe//Sitzungsberiehte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. Phvs.-math. 1896. P. 689-703.

11. Chebotarëv N.G. Opredelenie plotnosti sovokuponosti prostvkh chisel, prnadlezhashchikh zadannomu klassu podstanovok (determination of the density of the set of prime numbers belonging to a given substitution class)//Izv. Ross. Akad, Nauk. 1923. V. 17. P. 205-250.

12. Abdeljaouad I., Bouazizi F., Valibouze A. Certification algébrique pour le calcul de la résolvante de Lagrange. Archives HAL-CNRS, 2010. URL: http://hal,arehives-ouvertes.fr/hal-00483257/en/.

Accepted for edition 7.06.2010.

ПАРАЛЛЕЛЬНОЕ ВЫЧИСЛЕНИЕ РЕЗОЛЬВЕНТ ЛАГРАНЖА С ПОМОЩЬЮ МУЛЬТИРЕЗОЛЬВЕНТ © Филипп Обри

Университет Пьера и Мари Кюри, Париж, 75252, Франция, доктор наук, профессор,

e-mail: Philippe.Aubrv@upmc.fr

©

Университет Пьера и Мари Кюри, Париж, 75252, Франция, доктор наук, профессор,

e-mail: Annick.Valibouze@upmc.fr

Ключевые слова: резольвента Лагранжа; группа Галуа; идеал Галуа; компьютерная алгебра; параллельные вычисления.

Целью данной работы является создание параллельного алгоритма вычисления резольвенты Лагранжа для полинома одной переменной. Вычисление резольвенты Лагранжа для полинома одной переменной важно для теории Галуа. Начиная с алгоритма Лагранжа, было получено много других частных резольвент, называемых абсолютными, по основной теореме о симметрических функциях. Алгоритмов для не абсолютных резольвент мало и они получены недавно, так как они используют идеалы Галуа, которые были введены недавно. Эти алгоритмы с ростом степени полинома требуют больших затрат времени и памяти. Поэтому требуется распараллеливание. В 2004 году N. Rennert предложил модулярный алгоритм для вычисления абсолютных резольвент для целочисленных полиномов. Мы показываем, что его техника может быть применена для любых резольвент. Такой алгоритм естественно распараллеливается. Кроме того, мы предлагаем формулу для разложения резольвент, которая дает дополнительное распараллеливание. Тем самым мы получаем алгоритм с двумя уровнями распараллеливания.