Gr obner Duality and Multiplicities in Polynomial ...

Grobner Duality and Multiplicities in Polynomial System Solving  Maria Grazia Marinari Dipartimento di Matematica Universita di Genova [email protected]

Teo Mora y Dipartimento di Informatica Universita di Genova [email protected]

Abstract This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of nding the representations. We also discuss the current approach to the representation of real roots. 1 Introduction When solving a 0 - dimensional system of polynomial equations it is often satisfactory to know the zeroes of the system in such a way, that one can perform arithmetical operations on the coordinates of each root. One could however be interested also in the multiplicity of each root, not just in the weak \arithmetical" sense of simple, double, triple, etc. root, but in the stronger \algebraic" sense of giving a suitable description of the primary component at a root of the ideal de ning the solution set of the system. The aim of this paper is to discuss a representation of the \algebraic" multiplicity and ways of computing it. There is of course a stream of research about methods for solving systems of equations (for a survey we refer to [L93]) and also a "re ection" about the "meaning" of solving a system. The philosophy essentially goes back to Kronecker: a system is solved, if each root is represented in a way which allows to perform any arithmetical operations over the arithmetical expressions of its coordinates (the operations include, in the real case, numerical interpolation). For instance, in the classical Kronecker method, concerning the univariate case, one is given a tower of algebraic eld extensions over the eld of rational numbers, each eld being a polynomial ring over the previous one modulo the  This research was performed with the contribution of CEC Basic Research ESPRIT contract n. 6846 POSSO yThe coauthor thanks the GAGE, E cole Polytechnique, Palaiseau in Paris for hospitality and nancial support during the preparation of this paper

Hans Michael Moller Fachbereich Mathematik FernUniversitat Hagen [email protected]

ideal generated by a single polynomial and each root is represented by an element in such elds. The main eort of the actual research is devoted to eective techniques for representing roots of a systems and allowing ecient arithmetical operations over their expressions. In this setting of research, we are interested in how to describe a primary at a root of a system and how to be able to compute representation of such primary ideal, i.e. try to answer the question how to \compute multiplicity". The \arithmetical" multiplicity of a primary ideal for the maximal at the origin can be easily computed and read from the Hilbert polynomial but this doesn't give a sucient description of the \algebraic" multiplicity. In fact it is known that up to invertible transformations in K [[X;Y ]] there are exactly two classes of primaries at the origin having multiplicity 3, they are represented by (X 3 ; Y ) and (X; Y )2 . One could hope that more accurate invariants allow to distinguish primary ideals which are locally isomorphic, e.g. the Hilbert function (remark that of course it allows to discriminate the two classes above). However an example by Galligo [G] casts some doubts on this hope: it implies the existence of two primaries of the same multiplicity, which are not locally isomorphic and are not distinguishable by the Hilbert funcion and the Betti numbers. Therefore, in order to describe multiplicities of roots, we begin as a preliminary with the obvious \algebraic" approach which describes a root by studying the primary component of the ideal of the system corresponding to the given zero. While obviously Grobner bases and (mainly due to the \local" setting) standard bases are natural representations of primaries, a dierent approach was proposed by Grobner in order to reinterpret Macaulay notion of inverse systems. Grobner's suggestion is a generalization of the obvious univariate case, where  is a root of f (X ) of multiplicity d i @ nn (f )() = 0 8n; 0  n < d. @X Grobner proved that if a 2 kn is a common root of multivariate polynomials f1 ; : : : ; fr ; 2 k[X1; : : : ; Xn ] ( k any eld of characteristic 0 and k its algebraic closure), then there are nitely many linear combinations Di of partial deriva@ i1i +:::+in ; such that every polynomial f in the ideal tives @x i 1 1


@xnn (f1 ; : : : ; fr ) satis es Di (f )(a) = 0 for all i: Moreover the set of polynomials f satisfying Di (f )(a) = 0 for all i is exactly the primary component at a of the ideal (f1 ; : : : ; fr ). The Di are therefore a basis of a so called closed subspace of dierential conditions at the point. By the way, its dimension is the \arithmetical" multiplicity of the root. Because

of this (Grobner theory is presented in x3 and detailed in [MMM2]), it is natural to describe a primary by a closed subset of dierential conditions. In our paper [MMM2] we studied how to transform any such representation into another one and how to compute them to represent a primary of a root of a system. We give here a survey of our results related to the Grobner proposal, which we call Grobner Duality 1 In this paper we give a brief survey of the Grobner Duality and of the relevant results contained in [MMM2] and related results. In x3 we reintroduce the duality concept by Grobner which proved the duality between primary ideals at the origin with \closed" subspaces of dierential conditions. We propose a speci c representation of dual bases (x4) in order to have the same complexity for the representation of a primary ideal given by dierential conditions and by a Grobner-standard basis. Then we show how it can be used to test whether a polynomial belongs to a primary (x5). In order to compute the dual basis representing a primary we introduce the notion of continuation, which allows to nd all primaries containing the given one and with multiplicity 1 higher (x6) and then we introduce algorithms to compute continuations (x7). They allow to compute a dual basis of a primary given by any basis (Grobner, standard or whatever) (x8). We also discuss the current approaches in representing the roots of a system following Kronecker's philosophy and the important generalizations by Duval [D] (x9). On the basis of that, we show that applying the algorithm discussed in x8 to a 0-dimensional multivariate ideal and computing arithmetics \a la Duval" we get a sort of primary factorization, exactly as the Duval algorithm provided distinct-power-factorization for univariate polynomials (x10). 2 Preliminary and Notation 2.1 Notation Let k be an eective eld of characteristic zero, k its algebraic closure, P := k[X1 ; : : : ; Xn ], T the semigroup generated by fX1 ; : : : ; Xng, 8i  1 Ti the semigroup generated by fX1 ; : : : ; Xi g, m = (X1 ; : : : ; Xn ) the maximal ideal at the origin. Let K  k be a nite algebraic extension of k and let PK := K [X1; : : : ; Xn ]; for an ideal I  P , we will denote IK := IPK . P For a sparse polynomial f = i=1 ci i , ci 6= 0 8i, j 6= j for i 6= j , we will call staircase generated by f , the set (f ) := f 2 T :  divides i for some ig ([MT]). 2.2 Primary decomposition A zero-dimensional ideal I  P has a primary decomposition I = q1 \ : : : \ qt , where each qi is mi -primary for a maximal ideal mi , and mi 6= mj for i 6= j . Each maximal ideal corresponds to a set of k-conjugate zeroes of I , whose coordinates live in the nite algebraic extension Ki := P =mi of the eld k.

1 In site of the conjuring name, Grobner Duality is not directly related with the duality notions connected with polynomial systems introduced in [FGS], [C],(cf also [BK]) while however it would be interesting to understand if such a relation exists. Grobner theory can be found in many papers by him and in his book [Gr]

If mi is linear, mi = (X1 ? a1 ; : : : ; Xn ? an ), then it de nes a rational root of I , a = (a1 ; : : : ; an ) 2 kn . If m := mi is not linear, and K := P =m, then mPK has a decomposition into maximal ideals in PK , mPK = n1 \ : : :n\ nr , the nj 's are k-conjugate, linear, de ning roots bj 2 K , which are conjugate over k; moreover m = nj \ P 8j . As for the m-primary q = qi in the decomposition of I , qPK has a primary decomposition, qPK = p1 \ :: : \ pr , where pj is nj -primary, the pj 's are k-conjugate and q = pj \ P 8j . If m  P is a maximal ideal, K := P =m and q is an m-primary ideal, then the (arithmetical) multiplicity of q is mult(q) := dimk (P =q). If q is an m-primary component of a 0-dimensional ideal I , where the roots of m are b1 ; : : : ; br 2 K n , corresponding to primaries pi 2 PK , the multiplicity of I in bi is mult(pi ) = dimK (PK =pi ) = dimk (P =q)= dimk (K ). See for instance [Gr], pp. 168 { 178. 2.3 Set up Let us x a 0-dimensional ideal I  P and a root a = (a1 ; : : : ; an ) 2 kn of I . We will denote: - K = k(a1 ; : : : ; an ) the minimal algebraic extension of k containing a; - m = (X1 ? a1 ; : : : ; Xn ? an )  PK the linear maximal ideal whose root is a; - q  PK , the m-primary component of IPK . Up to a translation (Xi 7! Xi + ai ) we can assume ai = 0, i.e. m to be the maximal and q a primary at the origin, and we will do so in order to simplify notation. 2.4 Complexity In our complexity considerations, we are strictly working in an arithmetical setting and we just count the number of rational operations in k required by our algorithms. Even worse, we will simply count the number operations required over an algebraic extension K = k(a1 ; : : : ; an ). In terms of algorithms for root representations (x9), this complexity can be reduced to the previous one, by multiplying it by a constant, which will be evaluated in x10. It is clear, that under this very weak complexity model, the important question of coecient growth cannot be considered at all. We will measure complexity in terms of the following parameters: - n, the number of variables in P - t = dimk (K ) - s := mult(I ) - r := mult(q) - m the cardinality of the input basis of the ideal I -  the sum of the cardinalities of (fi ) for fi in the input basis of I . Remark that if I is given through a reduced Grobner basis, m  ns and  = O(ns2 ) . Since we may need to perform a translation, for the following complexity discussions, it is preliminary to know, what eect of a translation on a polynomial f = P  cis the i=1 i i . It is easy to see from Taylor formula (see (**) below) that the only terms with non-zero coecients are necessarily contained in (f ) and that computing f (X + a)

requires O() arithmetical operations in K and so O(t2 ) operations in k, where  is the number of terms in f ,  the cardinality of (f ). Moreover, if f is an element in the reduced Grobner basis of a 0-dimensional ideal I , then   s + 1, (f )  s, so, in case of translation, the space complexity of a basis of I is still bounded by2 3 = O(ns2 ) and the time complexity of computing it O(t ns ) . 3 Grobner Duality We denote by D(i1 ; : : : ; in ) : P ! P the dierential operator: i1 +


+in D(i1 ; : : : ; in ) = i ! : 1: : i ! @i1 in : 1

n

@X1


@Xn

This notation will be however simpli ed by denoting D(t) := D(i1 ; : : : ; in ) where t = X1i1 : : : Xnin . Also, i1 +


+ in = deg(t) will be called the order of D(t), ord(D(t)). We moreover denote ID := fD(t) : t 2 Tg and SpanK (ID) the P K -vector space generated by ID; the order of an element ci D(ti ) 2 SpanK (ID), with ci 6= 0 8i, is max(ord(D(ti))). In addition IDj := fD(0; : : : ; 0; ij ; : : : ; in ) : ij 6= 0g. For t = X1d1


Xndn and  = X1e1


Xnen we get by straightforward calculations

D(t)( ) =

n ?


t 1 if  = t1

if t does not divide  ?
?
?
where t := ed11P


denn . P Hence if f =  2T c  2 PKPand L =  2T b D( ) 2 SpanK (IDP), one has L(f )(0) =  2T c b . Remark that for f = i=1 ci ti , the Taylor formula asserts that for a = (a1 ; : : : ; an ) P f (X1 + a1 ; : : : ; Xn + an ) = D( )(f )(a) 0

 2T

=

P



 P

 2T i=1 jti

ci

?t
t i i(

 

a): ()

For each j = 1; : : : ; n, Xj : SpanK (ID) ! SpanK (ID) is the antiderivative with respect to Xj , i.e. the linear map s.t.: n (i1 ; : : : ; ij ? 1; : : : ; in ) if ij > 0; Xj (D(i1 ; : : : ; in )) = D 0 otherwise. Since Xj Xi = Xi Xj 8i; j , the linear map t is de ned in the obvious way for each t 2 T. Let us consider now, for each j = 1; : : : ; n, the linear map Xj : SpanK (ID) ! SpanK (ID) s.t.: Xj (D(i1 ; : : : ; in )) = D(i1 ; : : : ; ij + 1; : : : ; in ) Again t is de ned in the obvious way for each t 2 T. and the relation t t = Id holds for each t 2 T. To simplify notations, let us denote j := Xj , j := Xj , and j := j j . Then the following relations hold: n ij > 0; * j (D(i1 ; : : : ; in )) = 0D(i1 ; : : : ; in ) ifotherwise. * j nl (D(i1 ; : : : ; in )) = l j (D(i1 ; : : : ; in )) (i1 ; : : : ; ij ? 1; : : : ; il + 1; : : : ; in ) if ij > 0; = D 0 otherwise.

If < is a semigroup ordering on T, the induced ordering on ID satis es L1 < L2 ) t (L1 ) < t (L2 ) 8t 2 T; 8L1 ; L2 2 ID: With respect to this ordering we can speak of the leading term T (L) of L 2 SpanK (ID) P in a completely analogous way as for a polynomial: if L = ci Di with ci 6= 0, Di 2 ID, D1 > D2 >


, then T (L) = D1 . A K -vector subspace V  SpanK (ID) is closed if 8t 2 T, 8L 2 V , t (L) 2 V and if dimk (V ) is nite. Let

=(V ) := ff 2 PK : L(f )(0) = 0 8L 2 V g: If I  m de ne
(I ) := fL 2 SpanK (ID) : L(f )(0) = 0 8f 2 Ig: Theorem 3.1 m-primary ideals of PK and closed subspaces of SpanK (ID) are under a biunivocal correspondence. More exactly every closed subspace V  SpanK (ID) corresponds to the m-primary ideal =(V ) and every m-primary ideal q corresponds to the closed subspace
(q), so that q = =(
(q)) and V =
(=(V )). 2 Moreover dimK (
(q)) = mult(q); mult(=(V )) = dimK (V ): Proof [Gr] 174{178 4 Representation of dierential condition Theorem 3.1 provides a way of representing any m-primary ideals q, by giving a basis of the closed subspace
(q). There is however a major problem with this representation, that it can be space-exponential in mult(q). Example 4.1 Consider the ideal

q := (X1r ; X2 ? X1 ; : : : ; Xn ? X1 ): Then
(q) is generated byfL0 : : : Lr?1 g where L0 = D(1) = Id, L1 = D(X1 ) +


+ D(Xn), and in general Li =

X

t2T deg(t)=i

D(t); ?


so that an ideal of multiplicity r could require storing nr elements in K . There is an easy way out for this example: perform the change of coordinates Y1 := X1 , Y2 := X2 ? X1 , : : : , Yn := Xn ? X1 , so q = (Y1r ; Y2 ; : : : ; Yn ) and
(q) is now generated by fD(Y1i ) : i = 0 : : : r ? 1g. However this procedure doesn't apply to the next example. Example 4.2 Consider now the ideal I of the curve with parameteric equations X1 = t; X2r = t2 ; : : : ; Xn = tn and the m-primary ideal P qr := I + m . De ne w : T ! IN by w(Xi) := i, Li := D(t). It is possible to prove that t2T w(t)=i


(qr ) is generated by fL0 ; : : : ; Lr?1 g, requiring the storage of O((r=n)n ) K -elements and that this basis is the least space-consuming. It is therefore important to look for an alternative representation of closed subspaces which has less storage requirements. Here we propose one such representation.

2 We are generalizing the theorem to prove that the duality above relates ideals closed w.r.t. m-adic topology with vector subspace which are closed for antiderivative (but don't have nite dimension)

We start by discussing projections via dual bases. 0 So let Xi1 ; : : : ; Xid be a subset of variables and let P := k[Xi1 ; : : : ; Xi0d ], T0 the semigroup of terms of P 00, ID0 := fD(t) : t 2 T g and let  : SpanK (ID) ! SpanK (ID ) be the canonical projection. Proposition 4.3 Let q  P be a primary ideal, V :=
(q), q0 := q \ PK0 . Then (V ) =
(q0 ). Proof 0 First of all remark that (V ) is closed. Also, if f 2 PK and L 2 SpanK (ID), then L(f ) = (L)(f ), since D(t)(f ) =0 0 if t 62 T0 . Let L 2 (V ) and let 0L 2 V0 be s.t. (L) = L0 . L0 (f 0 ) 0= L(f 0 ) = 0 holds for each f 2 q . This implies (V ) 
(q ). Conversely, 0let 0 f 0 2 PK0 0 be 0s.t. L(f ) = 0 for 2 0 2allV L, so (V ). Then L ( f ) =  ( L )( f ) = 0 for all L f 2 q \ PK0 = q0 . This proves =((V ))  q0 . Since (V ) is closed, we obtain (V ) =
=((V )) 
(q0 ). Each element L 2 SpanK (ID n fIdg) can be uniquely written P as L = L1 +


Ln where Lj 2 SpanK (IDj ). Denote Lj := ni=j Li ; analogously, we will use also the notation Lj , L>j , L i - Li = i (Li ) = (i (L))i Proposition 4.5 Let U be a closed subspace of SpanK (ID), let ? := fL(1) ; : : : ; L(r) g be a basis of U . Let L be s.t. U +hLi is closed (i.e. in the terminology which will P be introduced in x6 : a continuation of U ). Then 8j , Lj = ri=1 cij j (L(i)j ) for some cij 2 K . Proof One has j (L) 2 U so j (L) = Pri=1 cij L(i) . Therefore:

Lj = j (Lj ) = j (j (Lj )) =

r X i=1

j (cij L(i)j ):

Corollary 4.6 A closed vector space V  SpanK (ID) with dim(V ) = r can be represented by O(nr2 ) elements in K . Proof Let ? := fL(1) ; : : : ; L(r) g be a basis of V , where, w.l.o.g. T (L(i) ) < T (L(j) ) for i < j: Hence L(1) = Id and moreover, as it is easy to prove, each hL(1) ; : : : ; L(`) i is closed. To represent L(`) , one can just represent each Lj(`) and it one has just to assign the coecients of P to represent (i) ). c  ( L ij j  j i Xn , it holds in particular that: - D(t) =  T (L() ) and (`) := minfj : 9i cij 6= 0g where  := (`) and  := maxfi : ci 6= 0g; - D(t) 2 hX ; : : : ; Xn i; D(t) 62 hX+1 ; : : : ; Xn i - L(i) = L(i) for all i with ci 6= 0; - L(`) 2 Span K (ID ):

Let now U be the closed vector space generated by

fL(1) ; : : : ; L(`?1) g; `) of
(q), one could try to proto nd the next generator L(P P

duce an element L = nj=1 i ts , D(ti ) 2 C (U ) such that L = CU;t0 + si=1 ci CU;ti . Proof 1) is satis ed if and only if L is a continuation of U . The assertion follows then from the easy remark that a continuation of U is a linear combination of elementary continuations.

7 Computing elementary continuations 7.1 Computing elementary continuations under the lexicographical ordering Let us now restrict to the case in which < is the lexicographical ordering and let us give a criterion to decide if the elementary continuation of U at t exists, which is easily transformed into an algorithm to compute it. Theorem 7.1 Let D(t) 2 C (U ) \ ID and let L(i ) 2 ? be s.t. T (L(i ) ) =  (D(t)). For   j  n let I (j ) be the set of the indices i s.t. - T (j (L(i) )) 62 T (U ), - T (j (L(i) )) < D(t) { so that L(i) 2 SpanK (ID ) - if T (j (L(i) )) 2 C (U ) then there is no elementary continuation of U at T (j (L(i) )) Then the following conditions are equivalent: 1) The elementary continuation C := CU;t exists 2) The following set of 1 + 2(n ? k) linear equations has solutions d(j;i) 2 K : P

C () =  (L(i ) ) + i2I () d(;i) (L(i) ) (c ) P (i) +1 (C () ) = (d+1 ) i2I (+1) d(+1;i) L P (i) C (+1) = i2I (+1) d(+1;i) +1 (L+1 ) (c+1 ) j (

j?1 X l=

.. .

P

C (l) ) = C (j) =

.. . C (n) =

(i) i2I (j) d(j;i) Lj?1

(dj )

P

(i) i2I (j) d(j;i) j (Lj )

(cj )

P

(i) i2I (n) d(n;i) n (Ln )

(cn )

Moreover, if the above conditions are satis ed, 8j  ; Cj = C (j) , while Cj = 0 for j < . Proof Let P C exist. Then  (C ) =  (C ) 2 U , so  (C ) = LP(i ) + i d(;i) L(i) for some d(;i) and C =  (L(i ) ) + (i) () i d(;i)  (L ) =: C . In the sum moreover, i is restricted to I (), since otherwise there would appear in C terms D( ) 2 C (U ), where an elementary continuation exists, a contradiction to the assumption that C is elementary. Assume now there are d(j; i), j < , i 2 I (j ) satisfying (c ), (d+1 ), : : :, (d?1 ), (c?1 ) and s.t. moreover C (j) = Cj 8j < . P?1 One has  (C ) = P  ( l= C (l) ) +  C 2 U . So there are d(;i) s.t.  (C ) = i d(;i) L(i) , which implies P?1

P

(i) i d(;i) L?1 ; P  (C ) = i d(j;i) L(i) , P C () := C = i d(;i)  (L(i) ).

 (

l=

C (l) ) =

The same argument as above shows that in the sum i is restricted to I (). Conversely assume that the given P system of linear equations has solutions d(j;i) and let C := C (j) , so that C (j) = Cj 8j . For each j one has

j (C ) = j ( = =

j?1 X l=

X

i2I (j) X

i2I (j)

C (l) ) + j (C (j) ) =

d(j;i) L(i)j?1 + d(j;i)

L(i)

X

i2I (j)

d(j;i) L(i)j =

so that j (C ) 2 U . Since for each (j;i), i 2 I (j ), the leading term of C is  (T (L(i ) )) = D(t) and no term D( ) in C (U ) appears in the expansion of C such that a continuation of U at  exists, nor a term D( ) 2 T (U ) appears in the expansion of C , then C is the elementary continuation of U at t. In order to solve the above system of equations with good complexity, one must however use the Gauss basis BV = fj (L(ji) )g. So, adopting freely the notation of Corollary 4.7, one also needs to know: - a representation of each L(i)j w.r.t. ?j

- 8j , 8 > j , 8i,  j (L(ji) ). In view of an iterative application of the P algorithm, this means also that for a continuation C = cji j (L(ji) ) one must be able to compute: a) a Gauss basis of Vj + hCj i b) a representation of  j (Cj ) w.r.t. BV 8j;  > j . To solve item a), let us remark that eachPL(ji) itself has a unique Gauss representation as L(ji) = d (L() ). Performing P Gaussian elimination on the Gauss representation Cj = c (L()) allows both to extend the basis ?j and to obtain a Gauss representation of the same kind also for the new basis element (if any). As for b), it is sucient to give a solution for the elementary continuation obtained by the algorithm P of Theorem 7.1; ()) from the Gaussian representation C = c  j P   (L one obtains  j (Cj ) = j c   (L() ) : By assumption one knows a representation of

L :=

X

c (L() )

in terms of BV , from which (since L 2PVj ) one obtains a representation in terms of ?j , L = di L(ji) so that P  j (Cj ) = di j (L(ji) ). We can now discuss the complexity of the algorithm outlined in Theorem 7.1, assuming the knowledge above. While the linear system of equations of Theorem 7.1 has a block structure which simpli es its solution, we will not take this into account in computing the complexity of solving it. We have therefore O(nr) unknowns d(j;i) imposing relations on the coecients of the representation of j (L(i) ) in terms of BV and so O(nr3) 3equations. The system can therefore be solved with O(n r ) arithmetical operations in K . Moreover each of the n auxiliary problems

- a) is a Gaussian elimination of a vector with (n ? j )r components over a subspace of dimension r; the total cost is therefore O(n2 r2 ) arithmetical operations in K . - b) it is again a set of Gaussian eliminations and so the complexity is the same. 7.2 Computing elementary continuations under any term ordering If < is not the lexicographical ordering, elementary continuations can still be computed by linear algebra, but the block structure of the equations coming from Theorem 7.1 is lost. We have however: Proposition 7.2 Let D(t) 2 C (U ) and let L() 2 ? be s.t. T (L() ) = D(t). Denote by J (j ), for 1  j  n, the set of the indices i s.t. a) T (j (L(ji) )) 62 T (U ), b) T (j (L(ji) )) < D(t) c) if T (j (L(ji) )) 2 C (U ) then there is no elementary continuation of U at T (j (L(i) )). The following conditions are equivalent: 1) The elementary continuation CU;t exists 2) The following set of linear equations

  (L() )+

n X X j=1 i2J (j)

c(ji)  j (L(ji) ) =

n X i=1

di L(1i)

for  = 2; : : : ; n has solutions c(ji) ; di 2 K : Moreover, if the above conditions are satis ed,

CU;t =  (L() ) +

n X X

j=1 i2J (j)

c(ji) j (L(ji) ):

(ji) j c Proof If CU;t =  (L() )+ Pnj=1 Pri=1 (ji) j (L ), then c(ji) = 0 unless i 2 J (j ) since in the expansion of CU;t there is no term in T (U ), nor terms in C (U ) having elementary continuations and moreover D(t) = T (CU;t ) =  T (L() ) > T (j L(ji) ) for each pair (j; i) s.t. c(ji) = 6 0. Moreover  ( C ) 2 U , so that there are d s.t.  ( C )=  U;t i  U;t Pn (1i) :

i=1 di L

Conversely let

C =  (L() ) + be such that  (C ) =

1 (C ) =

Pn

n X X j=1 i2J (j)

i=1 di L

X

i2J (1)

c(ji) j (L(ji) )

(1i) ;

since

c(1i) 1 (L(1i) ) 2 U;

then U + hC i is closed. Since the sum is restricted on J (j ), C is the continuation of U at t. The algorithm needs to solve O(n2 r) equations in O(nr) unknowns and so O(n4 r3 ) arithmetical operations in K .

8 An algorithm to compute a primary component at the origin of a 0-dim. ideal After this discussion on elementary continuation, we now present an algorithm s.t. given a basis ff1 ; : : : ; fm g of a 0dimensional ideal I  P vanishing at the origin allows to compute the closed vector space
(I ) =
(q) of dierential conditions de ning q, the primary component at the origin of I . The algorithm uses the following data: - ?i = fL(i1) ; : : : ; L(ii ) g a Gauss basis for the projection Ui of a closed space U of dierential conditions, i = 1; : : : ; n: -  := fCU; : D( ) < T (L() )g, (The algorithms outlined in x7 to compute CU; for each  2 C (U ) are used here.) - B := f 2 C (U ) : D( ) > T (L() )g. At initialization: ?i := fL(i1) g where L(i1) = Id,  := ;, B := fD(Xi ) : i = 1; : : : ; ng. At termination: ?1 is the reduced Gauss basis of
(I ),  := fCU; :  2 C (U )g, B := ;.

Repeat t := minB ( ) B := B n ftg If CU;t exists then If 9c 2 K 8j = P 1:::;m : CU;t (fj )(0) =

then

9 Representation of roots All over the paper, given a root a = (a1 ; : : : ; an ) 2 kn of an ideal I , we assumed it were the origin by means of the translation Xi 7! Xi + ai . Of course making this assumption we need to specify some points: 1) Obviously we must assume to be able to compute over the eld K = k(a1 ; : : : ; an ) 2) and apparently to precompute the roots of I ; 3) in our complexity analisys we computed the number of operations in K and we should at least be able to reduce them to operations in k. The aim of this section is to discuss these points, reducing them to a presentation of the current re ection about the meaning of solving systems. In particular we will recall brie y dierent ways of representing the roots of a zerodimensional ideal, and indicate how to perform arithmetical operations over arithmetical expressions of its roots. As a consequence of that, the points above will be disposed of. Informally speaking an elementary \arithmetical expression" over a root a = (a1 ; : : : ; an ) 2 kn of a zero-dimensional ideal I is either: CY 2 = D(Y 2 ) L(5)

CXY 2

L(6)

CU; 2 c CU; (fj )(0)

1 := 1 + 1, P L(11 ) := CU;t ? CU; 2 c CU; , ?1 := ?1 [ fL(11 ) g For j = 2 : : : n do L := Gaussreduction(L(1j1 ) ; ?j ); If L 6= 0 then j := j + 1, L(jj ) := L, ?j := ?j [ fL(jj ) g B := B [ fj (D(t)) : j = 1; : : : ; n; Xj t 2 C (U )g else  :=  [ fCU;t g

until B = ;

As an example we compute the dierential conditions of degree at most6 6 of the node, i.e. the primary at the origin q = I + m 2 K [X;Y ] where I = (Y 2 ? X 2 ? X 3 ), m = (X; Y ), so q = (f1; f2; f3), with

L(7)

L(8) L(9)

CXY 4

CY 5

(2) (3)

4

4 The ordering is the lex one with X < Y . The dierential conditions produced are: L(1) L(2)

= = CX 2 = L(3) = L(4) =

D(1) D(X ) D(X 2 ) D(Y )

D(XY )

D(Y 2 ) + D(X 2 ) D(XY 2 ) + D(X 3 )

D(XY 2 ) + D(X 3 ) + D(X 2 ) D(Y 3 ) + D(X 2 Y )

D(XY 3 ) + D(X 3 Y ) + D(X 2 Y )

D(Y 4 ) + D(X 2 Y 2 ) + D(X 4 ) + D(X 3 ) D(XY 4 ) + D(X 3 Y 2 ) + D(X 2 Y 2 ) + +D(X 5 ) + 2D(X 4 ) + D(X 4 ) = D(Y 5 ) + D(X 2 Y 3 ) + D(X 4 Y ) + D(X 3 Y )

Notes (1) The computation of Y (L(4) ) (and analogous computations) are

f1 = Y 2 ? X 2 ? X 3; f2 = X 5 Y; f3 = X 6 (see Figure 2).

= = = = = = =

(4) (5)

obtained by X (L(4) ) = X Y (L(2) ) = Y (L(1) ) = L(3) , the last but one equality is in the table itself. The computation of L := X Y (L(4) ) (and analogous computations) are obtained by L = Y X Y (L(2) ) = Y Y (L(1) ) = Y (L(3) ): The computation of CXY 2 is as follows: one computes X Y (L(4) ) = Y (L(3) ) = L(5) Y , from which (4) one deduces CXY 2 = Y (L(4) ) + X (L(5) ) X = Y (L ) + X (CX 2 ). One also has X Y (CXY 2 ) = X Y Y (L(4) ) + X Y X (CX 2 ) = = Y (CY 2 ) + Y (CX 2 ) = Y (L(5) ); since X Y (L(4) ) = Y (L(3) ) = CY 2 and X Y X (CX 2 ) = X (CX 2 ) X (L(8) ) = X Y (L(6) ) = Y X (L(6) ) = Y (L(5) ) + Y (L(2) ) = L(7) + L(4) The computation of CXY 4 gives Y (L(8) ) and so one has to compute X Y (L(8) ) = Y (L(7) ) + Y (L(4) ) = L(9) + L(6) from which one gets CXY 4 = Y (L(8) ) + X (L(X 9)) + X (L(X 6) )

Figure 1: The example Y (
) X Y (
)
(f1 )(0)
(f2 )(0)
(f3 )(0) 1 L(1) D(1) 0 0 0 0 0 0 X L(2) X (L(1) ) L(1) 0 Y (L(1) ) 0 0 0 X 2 CX 2 X (L(2) ) L(2) 0 Y (L(1) ) ?1 0 0 Y L(3) Y (L(1) ) 0 L(1) 0 0 0 0 (4) (2) (3)( 1 ) (2) (3) ( 2 ) XY L Y (L ) L L Y (L ) 0 0 0 Y 2 CY 2 Y (L(3) ) 0 L(3) 0 1 0 0 L(5) CY 2 + CX 2 L(2) L(3) Y (L(2) ) 0 0 0 2 (4) ( X 5) (5) (4) (5) XY CXY 2 Y (L ) + X (L ) L L Y (L ) ?1 0 0 (3) L(6) CXY 2 + CX 2 L(5) + L(2) L(4) Y (L(5) ) + Y (L(2) ) 0 0 0 Y 3 L(7) Y (L(5) ) L(4) L(5) Y (L(4) ) 0 0 0 3 (8) (6) (7) (4)( 4 ) (6) (7) (4) XY L Y (L ) L +L L Y L ) + Y L ) 0 0 0 Y 4 L(9) Y (L(7) ) + X (L(X 6) ) L(6) L(7) Y L(8) ) ?1 0 0 4 (8) ( X 9) ( X 6) (9) (6) (8) (9) (6) XY CXY 4 Y (L ) + X (L ) + X (L ) L + L L Y (L ) + Y (L ) ?1 1 0 (5) 5 (9) ( X 9) ( X 6) (8) (6) (9) Y CY 5 Y (L ) + X (L ) + X (L ) L + L L Y (CXY 4 )Y 0 0 ?1

t




X (
)

- the assigment of ai for some i; - the sum, dierence or the product of two arithmetical expressions, - the inverse of a non-zero arithmetical expression, and an \arithmetical operation" is either one of the four elementary operations (to which one could add extractions of p-th roots over elds of nite characteristic p) or testing whether an arithmetical expression is 0. Observe, however, that an \arithmetical expression" is not exactly an algebraic number in k[a1 ; : : : ; an ], it is rather a set of instructions in prescribed order which, applied to (a1 ; : : : ; an ) produce such an algebraic number, and these instructions could include \branching" ones, like

of IK , m = n \P the maximal ideal in P whose roots are a and its k-conjugates, q = p \ P the m-primary component of I . Moreover we will set t := ta := dimk (K ) = mult(m), the number of roots which are k-conjugate to a, v := va := mult(p); r := ra := tv = mult(q) and since we are interested in the cost of apsame computation over all roots of I , we set u := mult( I ), which is the number of dierent roots. If A is a set containing a single element from each set of k-conjugate roots of I , one has

if exp1 = 0 then exp2 := exp3 else exp2 = (exp1 )?1 :

Depending on the representation of roots, we get dierent complexities for the arithmetical operations.

In fact a likely scenario is one in which the same complex computation is to be performed over all roots of a 0-dimensional ideal, but, due to the dierent arithmetical behaviour of dierent roots, branchings occur and lead to totally dierent computations. Example 9.1 For instance one could ask whether the polynomial ga (Z ) = Z 3 + 3aZ 2 +4 12Z + 42a is squarefree, where a is any root of f (X ) = X ? 13X + 36, i.e. a = 2; 3. This requires computing gcd(ga ; ga0 ) and testing if it is constant. It is easy to verify that the remainder of the division of ga by g3a0 is (8 ? 2a2 )Z , so the remainder is 0 (and ga = (Z + a) ) if a = 2, while it is non-zero if a = 3 requiring an (obvious!) further polynomial division to nd that ga is squarefree. Our interest will be therefore to describe representations for the algebraic numbers which are obtained by evaluating an arithmetical expression in one or all the roots of a zero-dimensional ideal, and to evaluate the space complexity of such a representation and the time complexity for performing an arithmetical operation over two algebraic numbers so represented. Let us xn a 0-dimensional I  P and a root a = (a1 ; : : : ; an ) 2 k of I . We will denote K = k(a1 ; : : : ; an ) the minimal algebraic extension of k containing a, n = (X1 ? a1 ; : : : ; Xn ? an )  PK the linear maximal ideal whose root is a, p the n-primary component

u=

X

a2A

ta ; s =

X

a2A

ra =

X

a2A

ta va :

9.1 Representation by a tower of algebraic extensions The classical way to represent a is Kronecker's representing of K as a tower of simple algebraic extensions. Let Ki := k(a1 ; : : : ; ai ), with K0 = k and K = Kn, di := dimKi?1 (Ki ), i : k[X1 ; : : : ; Xn ] ?! Ki [Xi+1 ; : : : ; Xn ]. Then, for each i, there is a unique monic polynomial fi 2 K (X1; : : : ; Xi?1 )[Xi ] s.t. - i?1 (fi) is the minimal polynomial of ai over Ki?1 , so that Ki ' Ki?1[Xi ]=fi ; - deg Xj (fi ) < dj for j < i, degXi (fi) = di ;

- m = (f1 ; : : : ; fn ) (this last assertion is known as \Nulldimensionaler Primbasissatz"). As a k-vector space, K is identi able with the subspace of P whose basis is the set of terms N(d), d = (d1 ; : : : ; dnQ ), so to represent each element of K one needs to store t = di elements in k and the information needed to encode K (i.e. the fi0 s) requires storing O(nt) elements of k. This identi cation is extended to a eld isomorphism by de ning recursively product and inverse computation over SpanKi?1 (1; : : : ; Xidi ?1 ) by division-with-remainder and by

Bezout identity (i.e. by the half-extended euclidean algorithm). Both algorithms have a complexity of O(d2i ) arithmetic operations over Ki?1 so an arithmetic operation in K has a cost of O(t2 ) operations in k. In this representation an element is 0 if and only if it is represented as such. If b = (b1 ; : : : ; bn ) is a k-conjugate root of a, i.e. an other root of m, then k(b1 ; : : : ; bi ) ' Ki ; therefore in this model one needs to represent only one root P for each conjugacy O(nt) = O(nu) class for a total space requirement of a2A elements in k; to represent an arithmetic expression over each root of I one needs to represent it once Pfor each conjugacy class, for a total storage of u = ta elements a2A inPk and2 to perform an arithmetical operation one needs O(ta )  O(u2 ) operations in k. a2A Remark that given I , to be able to represent its roots in this model, one needs to perform apprimary decomposition of I or a prime decomposition of I and this requires the ability of factorizing univariate polynomials over simple algebraic extensions of k; while algorithms to do that are known, they can hardly be considered ecient. 9.2 Representation by a simple algebraic extension Let c = (c2; : : : ; cn ) 2 kn?1 and let Lc : P ?! P be the linear change of coordinates de ned by: Pn n 1 + j=2 cj Xj i =1 Lc (Xi ) = X Xi i>1 If (a1 ; : : : ; an )P is a root of I the corresponding root of Lc (I ) is then (a1 ? ni=2 ciai ; a2 ; : : : ; an ). By the (misnomed) P Primitive Element Theorem, denoting a(c) := a1 ? ni=2 ci ai , there is a Zariski open set U  kn?1 s.t. 8c 2 U , K = k[a(c) ] and two dierent roots of Lc (I ) have dierent rst coordinates. For such a c there are therefore a monic irreducible polynomial g1 2 k[X1 ] of degree t, and polynomials g2 ; : : : ; gn 2 k[X1 ] of degree less than t, s.t. - g1 is the minimal polynomial of a(c) , - 8i > 1, ai = gi (a(c) ) , - Lc(m) = (g1 ; X2 ? g2 ; : : : ; Xn ? gn ) (this last assertion is known as \nulldimensionaler allgemeiner Primbasissatz"). The eld K can be identi ed as in the tparagraph above with the vector space Spank (1; X1 ; : : : ; X1?1 ); the storage requirements are therefore still O(nt) for the eld, and t for an arithmetic expression and the cost of the arithmetics O(t2 ) (O(nu), u and O(u2 ) respectively over all the roots); remark however that the coecients of the gi 's are usually much larger than those of the fi's of the previous paragraph. On the other side, extracting a root of Lc(I ) requires now only polynomial factorization over k. 9.3 Representation by a squarefree triangular set In order to avoid the requirement for costly polynomial factorization, Duval [D] proposed her \dynamic-evaluation" model, which started a series of \weak" models for the arithmetics of algebraic numbers recently introduced. Let us rst describe how to represent a simple algebraic extension k[a] in this model and then we will see how to

represent towers. Let f 2 k[X1 ] be a squarefree polynomial of degree d s.t. f (a) = 0, let f = f1


fl be its factorization (introduced only for theoretical purposes, since the aim of this model is to avoid factorization at all); let ai be a root of fi and to x notation let us assume that a = a1 ; then by the Chinese Remainder Theorem one has:

k[X1 ]=f '

l M i=1

k[X1]=fi '

l M i=1

k[ai ]

so that, denoting by the canonical projection of k[X1 ]=f over k[a], each element of the latter eld can be (non u/nique/-ly) represented by any counterimage in k[X1 ]=f , requiring the storage of d elements in k. Since is a ring morphism the three ring operations over k[a] can be performed over Span k (1; X1 ; : : : ; X1d?1 ) as we have seen in the preceding models at a cost of O(d2 ) arithmetical operations in k. However, since k[X1 ]=f has zero divisors, testing for equality to zero and inverting an element of k[a] is no more evident. Example 9.2 Let us go back to the preceding example, 2where we were computing gcd(ga ; ga0 ) for ga (Z4 ) = Z 32 + 3aZ +12Z +4a, where a is any root of f (X ) = X ? 13X + 36, and let us perform it with coecients in k[X ]=f . The rst polynomial division requires only the ring arithmetics of k[X ]=f and produces ga (Z ) = 31 (Z + a)ga0 (Z ) + (8 ? 2a2 )Z The next division requires dividing ga0 by (8 ? 2a2 )Z , so that we rst need to know whether 8 ? 2a2 is zero or not, since: - if0 8 ? 2a02 = 0, then the euclidean algorithm is ended, gcd(ga ; ga ) = ga2 , whence ga = (Z + a)3 - if 8 ? 2a 6= 0, then, after inverting it, a further (obvious) division is needed (again requiring only ring operations in k[X ]=f ), which produces

ga0 (Z ) = (8 ? 2a2 )?1 (3Z + 6a)(8 ? 2a2 )Z + 12 whence one concludes gcd(ga ; ga0 ) = 1. Of course, the answer depends by which root of f a is: in fact if a = 2 then 8 ? 2a2 = 0, if a = 3 then 8 ? 2a2 6= 0.

The \pons asinorum" here is that there is no need to compute the roots of f in order to answer: in fact, since a is a root of f , h(a) = 0 if and only if a is a root of f (0) = gcd(f;h) while h(a) 6= 0 if a is a root of f (1) = f=f (0) , in which case its inverse can be computed by the half-extended Euclidean algorithm applied to h and f (1) . Example 9.2 (cont'd) In the above example one gets f (0) = gcd(X 4 ? 13X 2 + 36; 8 ? 2X 2 ) = X 2 ? 4, f (1) = f=f (0) = X 2 ? 9, nding a partial factorization of f with no recourse to a factorization algorithm. Anytime one needs to test if an expression h(a) is zero, one therefore has to perform the computation gcd(f; h) and if the result is non trivial one obtains a partial decomposition

k[X1]=f = k[X1 ]=f (0)  k[X1]=f (1) : The computation can then be continued on the summand of which a is a root (if this can be decided, say e.g. if a is the only real, or the only positive, root of f ) or separately on both summands.

Having discussed in detail Duval's model for a simple extension, let us discuss its multivariate generalization. In Duval's model a root a = (a1 ; : : : ; an ) is given if monic polynomials fi 2 k(X1 ; : : : ; Xi?1 )[Xi ] are given s.t. - 8(b1 ; : : : ; bi?1 ) root of (f1 ; : : : ; fi )  k[X1 ; : : : ; Xn ] the polynomial fi (b1 ; : : : ; bi?1 ; Xi ) is squarefree (a condition which can be tested by a gcd computation over k[b1 ; : : : ; bi ] so that the test can be inductively performed in this model) - degXi (fj ) < di 8j > i - fi(a1 ; : : : ; ai ) = 0 8i This allows to represent elements of k[a1 ; : : : ; an ] by elements of Spank (N(d)), d = (d1 ; : : : ; dn ) and to perform there ring operations. However anytime a zero-testing or an inversion is needed, this is performed by gcd computations over k(a1 ; : : : ; an?1 )[Xn] and this could lead to a splitting fn = fn(0) fn(1) . Remark that such gcd computations require the eld arithmetics of k(a1 ; : : : ; an?1 ), which is performed recursively in the representation k[X1 ; : : : ; Xn?1 ]=(f1 ; : : : ; fn?1 ) and so could itself, recursively, produce splitting at lower levels. A set of polynomials ff1 ; : : : ; fn g satisfying the conditions above is known as a triangular set; algorithms ([L89, M]) are known which given a (Grobner) basis of a zerodimensional ideal I allow to produce a family of triangular sets whose roots are disjoint and such that each root of I is a root of one of them. The sum of the k-dimensions of these triangular sets is therefore exactly u, so that representing the triangular sets requires storing O(nu) elements in k, representing an arithmetical expression of a root (or of all roots) requires storing u elements in k and performing an arithmetical operation over all roots O(u2 ) arithmetical operations in k. 9.4 Representation by the Shape Lemma As with the classical model, the Primitive Element Theorem allows to avoid recursion also in the model above; in fact an obvious generalization of it, the so-called Shape Lemma ([GM]) asserts that if I is a radical zero-dimensional ideal (and each ideal generated by a triangular set is such), then there is a Zariski open set U  kn?1 s.t. 8c 2 U , Lc (I ) = (g1 (X1 ); X2 ? g2 (X1 ); : : : ; Xn ? gn (X1 )) where g1 is monic and squarefree and deg(gi ) < deg(g1 ) 8i. This allows to represent k[a] a la Duval by k[X1 ]=f1 . Again, the advantage (if any) of avoiding recursive splitting is to be compensated by the larger coecients appearing in g1 and in the gi 's. The space and time complexity of this model are again O(nu); O(u); O(u2 ) respectively. 9.5 Representation by a simple squarefree extension At least the disadvantage represented by the size of the coef cients of the gi 's , i > 1, in the model above can be circumvented by a proposal contained in [ABRW]. The polynomial gi is just needed to Pgive a representation of ai as an expression in ac = a1 ? ni=2 ci ai . Any other representation of the form f=h where h; f 2 k[X1 ]=g1 and h is invertible is equally good. They show in0 particular that there are polynomials fi s.t. ai = fi (ac )=g1 (ac ) and that they usually have much

smaller coecients than the gi 's. On the one hand the coef cient operations are simpler by dealing with shorter coef cients, on the other hand the coecients0 are here rational numbers with denominators from S := fg1 (ac )i ji  0g: For getting the complexity, one needs a more detailed analysis. This is not yet done. 9.6 Representation by a radical border basis If I is a zero-dimensional ideal, the artinian algebra A = P =I is isomorphicLas a k-vector space to Spank (N(I )) and as a k-algebra to a2A P =qa where qa is the primary component of I whose roots are a and its conjugates. Therefore one has

p

p

Spank (N( I )) ' P = I '

M

a2A

P =ma '

M

a2A

k[a]

so that, exactly as in Duval'sp model, denoting the canonical morphism of Spank (N( I )) over k[a], each element of the latter eld can be (nonpuniquely) represented by any counterimage p in Spankp(N( I )), requiring the storage of card(N( I )) = mult( I ) = u elements in k. This representation has been proposed in [MT] where it is called the \natural" representation. p The multiplication in pP = I can be performed in the isomorphic copy Spanpk (N( I )) by Grobner p basis techniques: if f; g 2 Spank (N( I )), then Can(fg; I ) is pin the same residue class aspfg and belongs to Spank (N( I )); ifp the border basis p of pI is stored (i.e. the set f ? Can(; I ) : can be computed by  2 B( I )nN( I )g ), the product linear algebra techniques with O(u3 ) 2complexity; to store a border basis one needs to store O(nu ) elements in k. Inversion and zero-testing presents the same diculty that in Duval's model, and it is done by the ideal theoretic p generalization of gcd's: if h(X1 ; : : : ; Xn ) 2 Span k (N( I )), then: p p - h(a) = 0 if and only if a is a root of J (0) := I +( h); p - h(a) 6= 0 if andponly if a is a root of J (1) := I : h = ff 2 P : hf 2 Ig , p - P = I = P =J (0)  P =J (1) . A representation by border bases of both P =J (0) and P =J (1) can be computed by linear algebra techniques at a O(nu3 ) complexity. The exibility of this presentation (any Grobner basis can be used instead than the lexicographical one, implicitly used in Duval's model) and the absence of 2recursion, are compensated by a higher complexity: O(3nu ) to store the eld, O(u) to store a single element, O(nu ) for arithmetical operations. 9.7 Representation by a border basis With the \natural" representation, one is not restricted to work with radical ideals. In fact, similar to 2.6 one has Spank (N(I )) ' P =I '

M

a2A

 P =qa ?!

M

a2A

k[a];

with a projection : For polynomials h(X1 ; : : : ; Xn ) 2 Spank (N(I )), the following holds:

- h(a) = 0 if and only if a 2 A is a root of I + (h) - h(a) 6= 0 if and only if a 2 A is a root of I : h :=  ff 2 P : 9 h f 2 Ig: There is an integer s s.t. I : h = ff 2 P : hs f 2 Ig =: I : hs : Using I (0) := I + (hs )  I + (h) and I (1) := I : hs , one gets I = I (0) [ I (1) with Spank (N(I (0) ))  Span k (N(I (1))) '

' P =I (0)  P =I (1) ' P =I '

M

a2A

 P =qa ?!

M

a2A

k[a]

and multiplicities of the roots are preserved. The complexity of this model, whose interest seems to be in preservation of multiplicities, is therefore O(ns32 ) to store the eld, O(s) to store a single element, O(ns ) for arithmetical operations.

Summing up: complexity

Let us now summarize the complexity of representing the roots of an ideal in any of the models above as well as the complexity of representing an arithmetical expression and of performing an arithmetical operation. We have that the storage for representing all the roots of I is2 O(nu) for all the models except2 9.6, which requires O(nu ) and 9.7, which requires O(ns ); storing an arithmetical expression requires O(u) for all the models except 9.7, which requires O(s); performing an arithmetical opera2 ) for all the models except tion has a time complexity of O ( u 9.6, which requires O(u3 ) and 9.7, which requires O(s3 ).

Summing up: \solving" a system

According to the Kronecker - Duval phylosophy about solving systems of equation, which we have discussed above, a group R of roots of the ideal I is therefore given by giving - an artinian ring A - n elements 1; : : : ; n 2 A s.t. if we denote L - A = i=1 Ai the decomposition of A into irreducible algebras, - Ki the residue eld of Ai - i : A ?! Ki the canonical projection, - ai := ( i (1 ); : : : ; i (n)) 2 Kin we have R = fai : i = 1 : : :  g: Also, any algorithm to \solve" the ideal I is just required to returns a nite set of groups of roots R representing them as above; eg following x9.6, we could assume that \solving" I requires just to return a Grobner basis of I , so that A = P =I and i is the class of Xi ; in this way of course all the roots of I are represented by (1 ; : : : ; n ).

10 \Distinct power factorization" of multivariate polynomial 0-dim. ideals When Duval [D] introduced her model for roots of univariate polynmial, she strikng showed that applying her model to compute gcd (f (X ? );f 0 (X ? )) 2 k[] for each root  of f (X ) 2 k[X ] produced the distinct power factorization of f (X ). It is quite evident that according to the Kronecker-Duval approach to polynomial system solving, if we are given an ideal I = ff1 ; : : : ; fm g, and we 1) \solve" it in the sense above (i.e. we give an arthinian ring A and elements 1 ; : : : ; n 2 A for each grups of roots R 2) consider the ideal IA = I A[X1; : : : ; Xn ] and perform the translation Xj ! Xj ? j , so that the \generic" root aj 2 R is moved to the origin and so the primary component of IA relative to it is now a primary at the origin. 3) perform the algorithm described in x8 then as a consequence, we obtain - a split ofj the given set of roots as R = [Rj , each subset jR being represented as above by an artinian ring A and \roots" jk 2 Aj - closed subsets V j  SpanAj (ID) of dierential conditions, so that with the same notation as above for each root ajk 2 Rj , the subset ij (V j )  SpanKij (ID) is such that ij (V j ) =
(I ), where of course ij : Span Aj (ID) ?! SpanKij (ID) is the obvious extension of ij : Aj ?! Kij In other words, the algorithm described in x8 using the Kronecker-Duval approach to represent roots, is nothing more than the generalization to the multivariate case of the Duval algorithm to compute the distinct-power-factorization of a univariate polynomial. That this approach could be applied in a natural way to ideal factorization is so obvious that deserve at most this little remark 5 References [ABRW] M. E. Alonso, E. Becker, M.-F. Roy, T. Wormann, Zeroes, multiplicities and idempotents for zerodimensional systems, Proc. MEGA '94 (eds.: T. Recio and L. Gonzalez-Vega), to appear. [BK] S. Beck, M. Kreuzer How to compute the canonical module of a set of points Proc. MEGA '94 (eds.: T. Recio and L. Gonzalez-Vega), to appear. [C] J. P. Cardinal Dualite et algorithmes interactifs pour la resolution de systemess polynomiaux, These, Rennes I (1993) [D] D. Duval, Diverses questions relatives au calcul formel avec de nombres algebriques, These d'E tat, Grenoble 1987.

5 whose aim is to allow us to claim in this note that we hope to be able to apply more seriously Grobner Duality to the factorization problem

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