Multivariate Polynomial System Solving Using Intersections of Eigenspaces

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1 J. Symbolic Computation (2) 32, doi:.6/jsco Available online at on Multivariate Polynomial System Solving Using Intersections of Eigenspaces H. MICHAEL MÖLLER AND RALF TENBERG FB Mathematik der Universität Dortmund, 4422 Dortmund, Germany The solutions of a polynomial system can be computed using eigenvalues and eigenvectors of certain endomorphisms. There are two different approaches, one by using the (right) eigenvectors of the representation matrices, one by using the (right) eigenvectors of their transposed ones, i.e. their left eigenvectors. For both approaches, we describe the common eigenspaces and give an algorithm for computing the solution of the algebraic system. As a byproduct, we present a new method for computing radicals of zero-dimensional ideals. c 2 Academic Press. Introduction The problem of finding the common solutions of a system of polynomial equations, f (x,..., x n ) =, f s (x,..., x n ) =, or in algebraic terms, the problem of finding the variety V of the ideal A := (f,..., f s ), can be transformed into one or several eigenspace problems if the set of solutions is finite. See for instance the motivating note Corless (996) or the more detailed exposition in Cox et al. (998, Chap. 2, Section 4). It is well known that for n = s = the Frobenius companion matrix has the solutions as eigenvalues. For n >, there are classical results which describe for a fixed polynomial f of the given polynomial ring P invariants of the endomorphism Φ f : P/A P/A,. [g] [f g]. The set {f(y) y V } is the set of eigenvalues of Φ f. So these theorems describe just the image of V under f or parts of it, see for instance Gonzalez-Vega et al. (999). By an eigenspace method, we understand a method which uses one or several sets of eigenvalues f(v ) (and their eigenspaces) for the reconstruction of V. A naive method is as follows. One tests every combination (λ,..., λ n ), λ i eigenvalue of a representation matrix of Φ xi, for a common zero of f,..., f s. So each test requires s polynomial evaluations. This is reasonable if there are only a few different values λ i, see for instance p. 63 of Gonzalez-Vega et al. (999). In the worst case, if all λ i are different, one has card(v ) n tests for finding the card(v ) solutions, in a certain sense the search for m needles in a haystack of size m n. Using u-resultants, Lazard (98) considered methods for combining eigenvalues of two endomorphisms Φ xi and Φ xj in order to reduce the haystack investigation. A result // $35./ c 2 Academic Press

2 54 H. M. Möller and R. Tenberg of Schur shows that there is a basis of P/A such that the representation matrices of all Φ xi, i =,..., n, are upper triangular. Denoting by α (j) i the jth diagonal element of the representation matrix to Φ xi, Yokoyama et al. (992) proved that the points α (j) := (α (j),..., α(j) n ) are the solutions of the given system of equations. A totally different way was proposed first by Auzinger and Stetter (988). Let M f denote the transpose of a representation matrix corresponding to Φ f. M f is called the multiplication matrix. Then for every z = (z,..., z n ) V there is a right eigenvector v to the eigenvalue f(z), M f v = f(z)v, () v independent of f, which allows one to read off all components z i of z. If all eigenspaces of M f have dimension, in other words if M f is non-derogatory, then such v is easily detected. Whereas in Auzinger and Stetter (988) resultant methods were used, the method was described and extended in Möller and Stetter (995) using Gröbner techniques. In that paper, procedures were also proposed on how to reconstruct the points of V. A discussion on the stability of such procedures can be found in several papers of Stetter, for instance Stetter (996). Another method which is based on a Schur factorization of multiplication matrices was proposed by Corless et al. (997). They also consider stability aspects of their procedure. There are also other eigenspace methods for special instances like Manocha s for the complete intersection case, which is presented in Cox et al. (998, Chap. 3, Section 6), as an application of u-resultants. In this paper, we discuss the eigenspaces for both the representation matrices and the multiplication matrices. For this purpose, we briefly present the necessary definitions from Gröbner techniques and (for a better understanding of multiple points) the concept of dual bases. The investigation of common eigenspaces of the representation matrices leads to a new and simple calculation of the radical A of a zero-dimensional ideal A as A = A : q, q being a polynomial. We extend the subspace method that was already indicated in Möller and Stetter (995) to a method for computing the variety V starting with either the representation matrices or the multiplication matrices for the endomorphisms Φ x,..., Φ xn. We stress, explicitly, that our method is direct. It needs only a (numerical) method for computing the eigenspaces of a matrix. We need no points or coordinates in general position, which often means a finite number of trials until they are found. (Remember that you always have only a finite number of trials!) On the contrary, if we have not yet acquired sufficient information to reconstruct the vector v in () then we do not discard them but analyse them more carefully until we obtain v. This reuse of information is, in a sense, the ecologically correct procedure. 2. Basic Definitions and Results Gröbner bases are introduced in most textbooks on Computer Algebra. Unfortunately, notations differ slightly. Since one of our motivations is to clear misunderstandings about the eigenvalue method, we begin by fixing the basic notations. Let P denote the ring of polynomials in the variables x,..., x n with complex coefficients, i.e. P = C[x,..., x n ]. Definition. (Terms, Ordering) Let the set of terms T := {x i xin n i,..., i n N }

3 be ordered by an admissible ordering < T, i.e. (a) < T t for all t T \ {}, (b) if t < T t 2, then t t < T t 2 t for all t T. Multivariate Polynomial System Solving 55 The admissible term ordering is called a degree ordering, if deg(t ) < deg(t 2 ) implies t < T t 2 for arbitrary terms t, t 2. Having fixed an admissible term ordering, then every f P \ {} has in its representation as a linear combination of terms a maximal term w.r.t. < T, which is called the leading term lt(f) = lt <T (f). (We omit in the following the dependence of lt on < T.) In our definition, we followed B. Buchberger, who introduced the name Gröbner bases and many related notations. For us, a monomial is a term multiplied with an element from the ground field representing in a sense the singular where polynomial is the plural. Definition 2. (Normal Set, Normal Form) Let A be an ideal and < T an admissible term ordering. The set N of all terms not contained in the multiplicative semigroup {lt(f) f A} is called the normal set of A. The projection NF : P span N orthogonal to A, i.e. NF(f) f A for all f P, NF(f) = f for all f span N, is called the normal form mapping, and NF(f) the normal form of f. Instead of the notation normal set the names standard monomials and basis monomials are often used in literature. We prefer the notation normal set because it reminds one of the normal form mapping NF. The normal set, and hence the normal form, depends on the ideal and (by the leading terms) on the term ordering. Definition 3. (Gröbner Basis) Let A P be an ideal, and < T an admissible ordering. The set G = {g,..., g s } A is called a Gröbner basis if for every f A there exist a g G such that lt(g) divides lt(f). From a Gröbner basis G of the ideal A w.r.t. the term order < T the associated normal set can be computed as N = {t T g G s.t. lt(g) divides t}. Gröbner bases also allow the efficient computation of the normal form NF(f) of f, see for instance Cox et al. (998). For a fixed ideal A, P/A = {[f] f P}, [f] = {g f g A} is an algebra. We consider it as a vector space. Then, as stated in Möller and Stetter (995), the following holds. Proposition. If N is the normal set of the ideal A w.r.t. an arbitrary admissible term ordering, then {[t] t N } is a linear basis of the vector space P/A. Let then [f] = t N c t(f)[t]. NF(f) = t N c t (f)t,

4 56 H. M. Möller and R. Tenberg Proof. f NF(f) A. Hence [f] = [NF(f)]. By linearity of the mapping g [g], {[t] t N } generates P/A. It is also a basis because [ ] c t [t] = c t t = h := c t t A, t N t N t N and h leads to the contradiction lt(h) N by h span N and lt(h) N by h A. Definition 4. (Multiplication Matrix) Let A be an ideal and N = {t,..., t r } its normal set w.r.t. a given term order < T. If r [f t i ] = m ij (f)[t j ], i =,..., r j= then the complex r r matrix M f = (m ij (f)) r i,j= is called the multiplication matrix corresponding to f P. If we consider the linear endomorphism Φ f : P/A P/A, [g] [f g], and look at the representation matrix B f w.r.t. the basis which consists of the elements of the normal set, then we easily see that B f = M T f. Remark. This intimate connection of the multiplication matrix M f and the representation matrix B f led to confusion in previous papers on the eigenvalue method. Sometimes both were called multiplication matrices or tables. Left eigenvectors of one matrix are right eigenvectors of the transposed one. Since we want to study both systems of eigenvectors, and right and left depend on which of the matrices we prefer, we hope to clarify our representation by formulating results only for one type of eigenvectors of the respective matrix. For an r r matrix A we denote by E(λ, A) := {v C r Av = λv} the eigenspace to the eigenvalue λ of A. Remark 2. The main idea of the method proposed in this paper is to find specific vectors which are eigenvectors of all M f, f P, resp. B f, f P. Anticipating a result of Theorem 2, the eigenvalues of M f, and hence also of B f = Mf T, are of type f(), ranging over the common zeros of A, we have to compute E(f( ), M f ) and E(f( ), B f ). f P This infinite intersection can be avoided because the commuting families {M f f P} resp. {B f f P} are generated by {M x,..., M xn } resp. {B x,..., B xn }, n E(f( ), M f ) = E(i, M xi ), where = (,..., n ). f P f P i= f P n E(f( ), B f ) = E(i, B xi ), i=

5 3. Dual Bases Multivariate Polynomial System Solving 57 The multiplicity of a common zero z of a set of polynomials F is a useful piece of information. If one is interested in the geometric interpretation, how the algebraic surfaces f =, f F, intersect in z, one needs more than a proper number. Using the concept of dual bases which is investigated in Marinari et al. (993, 996) and which dates back to Lasker (95), Macaulay (96) and, especially, Gröbner (97), one gets a better insight into the geometric configuration at the intersection points. In addition, and this is why we introduce this concept here, it enables us to describe eigenspaces and to explain the decomposition of the multiplication matrices into a kind of Jordan normal form following Möller and Stetter (995). Definition 5. (Variety, Zero-dimensional) Let A be an ideal. The set V (A) = {z C n f(z) = f A} is called the variety of A. An ideal A is called zero-dimensional if its variety consists of a finite set of points. By the finiteness theorem (Cox et al., 998), A is zero-dimensional if and only if its normal set is finite. Hence exactly for zero-dimensional ideals finite multiplication matrices and representation matrices can be defined. By Hom(P, C) we denote the set of all linear complex valued functionals on P. Definition 6. (Dual Basis) Let V P be a vector space. If {L (),..., L (s) } is a basis of the vector space then it is called a dual basis of V. V := {L Hom(P, C) L(f) = f V } It is a nice linear algebra exercise to show that exactly the vector spaces V of finite codimension possess a dual basis and under which additional conditions on the dual basis V is an ideal. Here we concentrate on functionals of special type. Definition 7. (Differential Operators, Shifts) For terms t = x i xin n T we define the elementary differential operators D(t) : P P, p i! i n! i+ +in p x i xin n Finite linear combinations of these elementary operators will be called differential operators. For u T the shift σ u of the elementary differential operator D(t) is defined to be D( t u ) if u divides t and otherwise. By linearity the definition of shifts is extended to arbitrary differential operators. Obviously, the set of all non-zero shifts of an elementary differential operator D(t) is finite. Hence the same holds for arbitrary differential operators. Let L be an arbitrary differential operator. Then a non-zero constant multiple of D(), the identity operator, is among its shifts because L is a linear combination of elementary operators D(t). If D(u) is one of them with maximal differentiation order, then σ u L = cd() with a constant c..

6 58 H. M. Möller and R. Tenberg Given a finite dimensional subspace of span{d(t) t T } and a point z C n, then the set of all f P with L(f)(z) = for all L is a vector space. We need an additional condition on to ensure that this vector space is an ideal. Definition 8. (Closed Subspace) A linear subspace of span{d(t) t T } is called closed if its dimension is finite, and if L σ xi L, i =,..., n. Hence also σ u L if L and u T. Especially, every non-zero closed subspace contains D(), the identity operator. We remark that in Möller and Stetter (995) closed subspaces were by definition non-zero spaces. Here we modify the definition to avoid case distinctions in the theorems of Section 5. For z C n and L span{d(t) t T } we define the functional L z : P C, L z (f) := (Lf)(z), that is point evaluation of the polynomial Lf in z. For convenience, whenever we use the notation L z, z C n, for a functional, then we implicitly mean that L is a differential operator (followed by a point evaluation at z). The correspondence of dual bases of zero-dimensional ideals and closed subspaces of differential operators is given by the next theorem. Theorem. If A is a zero-dimensional ideal with V (A) = {z,..., z m }, then there are closed vector spaces of differential operators (k), such that f A L zk (f) = for all L (k), k =,..., m. Let {L (,k),..., L (s k,k) } denote a basis of (k), k =,..., m, then the s + + s m functionals L (j,k) : f (L (j,k) f)( ) are linearly independent and constitute a dual basis of the ideal A. Proof. (Marinari et al., 993) Conditions equivalent to L zk (f) = L (k) like L (i,k) (f) =, i =,..., s k, are called Max Noether conditions, (see Lasker, 95). They more precisely, the differential operators L (k) allow one to understand better how the manifolds p =, p A, intersect in. For instance by considering the differential operators of order one finds all tangents in in common to the manifolds. (These manifolds pass through since D() (k).) These Max Noether conditions give more insight than the proper number s k = dim (k), the multiplicity of as point of the variety of A. Therefore we will call (k) the Max Noether space of A at. The space (k) := {L zk L (k) } is often called the local dual space of A at. Having a basis {L (),..., L (s k) } of the Max Noether space of A at, then {L (),..., L (s k) } is a basis of (k) and hence a specific local dual basis (of A at ). An arbitrary local dual basis does not necessarily give as much geometrical information as a Max Noether space, because the local dual basis consists of functionals which may also be given as, say, definite integrals.

7 Multivariate Polynomial System Solving 59 The ideal Q k := {p P L zk (p) = L (k) } is a primary ideal with V (Q k ) = { }. By construction Q k = (k). As an easy consequence of Theorem, we get m A = Q k. k= Hence the dual basis of A as given in Theorem reflects the primary decomposition of A. In Möller (993) the definition of a consistently ordered basis was introduced. This is an ordered list {L (),..., L (s) } of differential operators, constituting a basis of a closed subspace, such that for every degree l > there is a j such that {L (),..., L (j) } is a basis of {L deg L < l}. Such a basis can always be computed by starting with L () := D() and enlarging a basis of {L deg L < l} to a basis of {L deg L l} for increasing l. The first element in a consistently ordered basis is always c D(), c a constant. For convenience, we always assume D() to be the first element if the basis is consistently ordered. A frequently used formula is the application of a differential operator on a product of two polynomials. For L span{d(t) t T } and f, g P the Leibniz rule gives L(f g) = u T D(u)f σ u L(g). (2) 4. Eigenspaces of M f Theorem 2. Let A be a zero-dimensional ideal, V (A) = {z,..., z m }, N = {t,..., t r } its normal set, and M f the multiplication matrix corresponding to f P. Then f(z ),..., f(z m ) are the eigenvalues of M f. Let (k) be the Max Noether space of A at, k =,..., m. If {L (,k),..., L (s k,k) } is a consistently ordered basis of (k), then the vectors satisfy M f D (,k) M f D (j,k) D (j,k) := (L (j,k) (t ), L (j,k) (t 2 ),..., L (j,k) (t r )) T C r f( ) D (,k) =, (3) f( ) D (j,k) = u T \{} D(u) zk f (σ u D (j,k) ) zk, j = 2,..., s k. (4) Proof. (Möller and Stetter, 995) In this theorem, the vector (σ u D (j,k) ) zk stands for ((σ u L (j,k) ) zk (t ), (σ u L (j,k) ) zk (t 2 ),..., (σ u L (j,k) ) zk (t r )) T. By the consistent ordering of the basis {L (,k),..., L (s k,k) }, the shifted operator σ u L (j,k), u T \{}, is a linear combination of operators L (i,k), i < j. Therefore, the identities (3) and (4) of Theorem 2 give, in matrix vector notation, the following. Corollary. Let A, V (A), and the vectors D z (j,k) k be as in Theorem 2. Define the matrix S := (D z (,),..., D z (s,), D z (,2) 2,..., D z (sm,m) m ). Then for arbitrary f P M f S = S J f,

8 52 H. M. Möller and R. Tenberg where J f is a block diagonal matrix and its kth diagonal block is an s k s k upper triangular matrix with diagonal entries f( ). All D z (,k) k are eigenvectors to the eigenvalue f( ) by (3). Hence, using Remark 2, n E(f( ), M f ) = E(i, M xi ). (5) D (,k) f P We will now prove that the vectors D z (,k) k, k =,..., m, (and their scalar multiples) are the only common eigenvectors. This result is also proved in Mourrain (998). Theorem 3. Let A, V (A), M f, and the vectors D z (j,k) k be as in Theorem 2. Then dim E(f( ), M f ) = and therefore f P f P i= E(f( ), M f ) = span{d (,k) }. Proof. We assume for simplicity that A is primary with V (A) = {z}. Let N = {t,..., t r } be its normal set. As Theorem 2 shows, every eigenvector to eigenvalue f(z) of M f is of type (L z (t ),..., L z (t r )) T, L denoting an element of the Max Noether space of A at z. For i =,..., n let f := x i. Then on the one hand M xi L z (t ). = z i L z (t r ) L z (t ). L z (t r ), (6) where f(z) = z i. On the other hand by the Leibniz rule L(x i t j ) = x i L(t j ) + (σ xi L)(t j ) one gets L z (x i t ) L z (t ) (σ xi L) z (t ). = z i. +.. (7) L z (x i t r ) L z (t r ) (σ xi L) z (t r ) The left-hand sides of (6) and (7) are equal since M xi is the multiplication table. Hence comparing the right-hand sides one gets (σ xi L) z (t j ) = for j =,..., r. (σ xi L) z A because L is in the Max Noether space and, by closedness, σ xi L is also. Therefore (σ xi L) z =. This holds only if σ xi L =. Since i was arbitrary, we obtain σ u L = for arbitrary shifts. This implies L = c D(), c a constant. The vector D (,k) = (L (,k) (t ), L (,k) (t 2 ),..., L (,k) (t r )) T allows one to read off the components of the point, since L (,k) is the identity D() and the variables x,..., x n are contained in the normal set {t,..., t r } or if an x i is not contained, its value at

9 Multivariate Polynomial System Solving 52 can easily be reconstructed by evaluating the Gröbner basis element g with lt(g) = x i at, (cf. Möller and Stetter, 995). Example. We consider the ideal A given by the polynomials g (x, y) = 2y 2 + x 2 2y, g 2 (x, y) = xy x, g 3 (x, y) = x 3. The set G = {g, g 2, g 3 } is a Gröbner basis w.r.t. a degree ordering with x < y and we get N = {, x, y, x 2 }. The variety of A is V (A) = {(, ), (, )}. We can use the algorithm described in Marinari et al. (996) to compute bases for the Max Noether spaces. It results in {D()} at z = (, ), {D(), D(x), D(x 2 ) 2 D(y) } at z 2 = (, ). Hence the dual basis by Theorem { D() (,), D() (,), D(x) (,), (D(x 2 ) } 2 D(y)) (,). The multiplication matrices w.r.t. x and y are easily set up: M x :=, M y :=. 2 Here, the matrix S of Corollary is (using L for D(x 2 ) 2 D(y)) D() (,) () D() (,) () D(x) (,) () L (,) () D() S := (,) (x) D() (,) (x) D(x) (,) (x) L (,) (x) = D() (,) (y) D() (,) (y) D(x) (,) (y) L (,) (y) D() (,) (x 2 ) D() (,) (x 2 ) D(x) (,) (x 2 ) L (,) (x 2 2 ) This gives M x S = SJ x and M y S = SJ y with the Jordan like matrices J x :=, J y := 2 The first two columns of S are the eigenvectors (5). Their second and third components are the x- and y- coordinates of the variety points (, ) and (, ).. 5. Eigenspaces of B f For the beginning, let us repeat a definition from eigenvalue theory. Definition 9. (Principal Vector) Let A be a r r matrix with eigenvalue λ. An eigenvector to eigenvalue λ is called a principal vector of first order. If v is a principal

10 522 H. M. Möller and R. Tenberg vector of order k to eigenvalue λ, then a vector w satisfying Aw = λw + v is called a principal vector (to eigenvalue λ) of order k +. In the following, we fix a polynomial f and a zero-dimensional ideal A P with variety V (A) = {z,..., z m } and normal set N = {t,..., t r }. Theorem 4. Let B f = M T f be the representation matrix of the endomorphism Φ f : P/A P/A, [g] [f g] w.r.t. the basis {[t ],..., [t r ]}. For every positive integer k the following statements are equivalent (a) b := (b,..., b r ) T is a principal vector of order k to eigenvalue λ of B f. (b) r i= b it i A : (f λ) k, but r i= b it i A : (f λ) k. Proof. Induction on k. Let k =. Then B f b = λb r b i t i f λ i= r b i t i A i= r b i t i A : (f λ). In addition r i= b it i A = A : () = A : (f λ) is equivalent to b. k k : Let p := r i= b it i A : (f λ) k, but p A : (f λ) k. Then q := p (f λ) A : (f λ) k, but not in A : (f λ) k 2. Let q =: g i t i + a with a A. g := (g,..., g r ) T is a principal vector of order k. By definition q = p (f λ). Hence B f b λb = g, i.e. b is a principal vector of order k. (a) (b) follows by reversing the conclusions. By (3) of Theorem 2 and Mf T = B f the eigenvalues of B f are also {f(z) z V (A)}. Hence for every z V (A), the polynomials having principal vectors of B f as coefficient vectors can be found in the ascending chain A A : (f f(z)) A : (f f(z)) 2 A : (f f(z)) r = A : (f f(z)) r+ =... which becomes stationary, because P is Noetherian. A : (f f(z)) j contains all polynomials corresponding to principal vectors of order at most j. Theorem 5. Let {L (,k),..., L (sk,k) } be a basis of the Max Noether space of A at, k =,..., m. Then the Max Noether space (j) of n i= A : (x i i ) at z j is generated by n {σ xi L (,k),..., σ xi L (sk,k) } if j = k, i= i= {L (,j),..., L (sj,j) } if j k. Proof. Since D(t)(x i i ) is for t = x i and for t T \ {, x i }, the Leibniz rule (2) gives L (l,j) ((x i i ) g) = (x i i )L (l,j) (g) + σ xi L (l,j) (g).

11 Now g n i= A : (x i i ) is equivalent to = L (l,j) z j Multivariate Polynomial System Solving 523 ((x i i ) g) = (z ji i )L z (l,j) j (g) + (σ xi L (l,j) ) zj (g) for l =,..., s j, j =,..., m, i =,..., n. If j k, then z ji i for an i {,..., n}. Hence using that σ xi L always has a smaller (differentiation) order than L and inspecting for j k the L (l,j) in increasing order, we see that for j k the space (j) is generated by L (l,j), l =,..., s j. And for j = k the space (j) is generated by σ xi L (l,j), i =,..., n, l =,..., s j. A Max Noether space at a point z is the null space, if and only if z does not belong to the variety. This happens in the theorem, iff is a simple zero of A (then local dual basis is exactly {D() zk }). Such is not in the variety of n i= A : (x i i ). If we know that n g A : (x i i ), g A, (8) i= without knowing all components i of explicitly, then we can reconstruct them because the coefficient vector of g is an eigenvector to eigenvalue i of B xi by Theorem 4. Example 2. We consider the same ideal A = (g, g 2, g 3 ) as at the end of the previous section. The representation matrices w.r.t. x and y are B x = Mx T =, B y = My T =. 2 By means of the matrices 2 T x :=, T y := 2 2 we get the Jordan normal forms B x T x = T x, B yt y = T y In T x the first two columns are first order principal vectors (eigenvectors). They are coefficient vectors of y and x 2 resp. Hence by Theorem 4 A : x = (g, g 2, g 3, y, x 2 ) = ( y, x 2 ). In T y the first column is an eigenvector to eigenvalue, hence A : y = (g, g 2, g 3, x 2 + 2y 2) = (g, g 2, x 2 + 2y 2), and second and fourth columns of T y are eigenvectors to giving, by Theorem 4, A : ( y) = (g, g 2, g 3, x 2, x) = (y 2 y, x)..

12 524 H. M. Möller and R. Tenberg By Theorem 5, we get the following bases of the Max Noether spaces of the quotients A : x A : y : at (, ), {D(), D(x), D(x 2 ) 2D(y)} at (, ), A : x A : ( y) : {D()} at (, ), {D(), D(x)} at (, ). 6. Generators of Max Noether Spaces Definition. Let A be zero-dimensional and z V (A). We say, the Max Noether space of A at z is generated by exactly k differential operators L (),..., L (k), if = span{σ u L (j) u T, j =,..., k}, and if less than k elements of generate only proper subspaces of. In the univariate case, every Max Noether space is generated by one element since closed subsets are for n = of type span{d(),..., D(x s )}, as already remarked in Möller and Stetter (995). We also remark that Max Noether spaces generated by one element are well known and studied deeply. Macaulay (96) called them principal systems, which are nowadays connected with Gorenstein rings (Eisenbud, 994, p. 529f). If a Max Noether space of a primary ideal Q at (the single zero) z is generated by L, then Q is irreducible, i.e. for any primaries Q, Q 2 with Q = Q Q 2 either Q = Q or Q 2 = Q holds. The reason is that V (Q ) = V (Q 2 ) = V (Q) by Q = Q Q 2, and L is either in the Max Noether space of Q or in that one of Q 2. But then this space contains, which means Q or Q 2 is a subideal of Q. Since Q = Q Q 2 it cannot be a proper subideal, see also Marinari et al. (993). If we take a differential operator and all its shifts, then the space generated by them is closed. Hence it is a Max Noether space of an irreducible primary ideal. We can construct in this way, for each of the k differential operators, a Max Noether space of a primary ideal. Thus we easily see that, if a Max Noether space of a primary ideal Q with variety {z} and generated by exactly k differential operators is given, then Q is the intersection of exactly k different irreducible primary ideals. Theorem 6. Let A be a zero-dimensional ideal and z V (A). The Max Noether space of A at z is generated by exactly k elements if and only if dim E(f(z), B f ) = k. f P Proof. For simplicity, we may assume V (A) = {z}, i.e. A is a primary ideal. Let be generated by k differential operators. These k operators together with their shifts generate as a vector space. Hence by cancelling linearly dependent elements, a basis {L (),..., L (s) } is obtained containing the k generators. Let L (j),..., L (jk) denote these generators. Take q,..., q s span N biorthogonal to the local dual basis {L () z,..., L (s) z }, i.e. L (i) z (q j ) = δ ij. Then by biorthogonality (σ u L (ji) ) z (q ji ) = for all proper shifts. Hence

13 the Leibniz rule (2) simplifies for arbitrary polynomials f to Multivariate Polynomial System Solving 525 L (j) z (f q ji ) = f(z)l (j) z (q ji ), j =,..., s. This means (f f(z))q ji is annihilated by L () z,..., L (s) z. Therefore (f f(z))q ji A, or in other words q ji A : (f f(z)), meaning by Theorem 4 that the coefficient vector of q ji is an eigenvector of B f for all f P. Since the q j,..., q jk are linearly independent, every eigenspace E(f(z), B f ) has at least dimension k. Assume, there is a vector v f P E(f(z), B f ) which is not a linear combination of the coefficient vectors of the polynomials q j,..., q jk. Let v be the coefficient vector of q span N. We may replace q by q := q k z ( q)q ji. Then L (ji) z (q) =, i =,..., k. The Leibniz rule gives for all f P i= L(ji) L (i) z (f q) = f(z)l (i) z (q) + D(u) z (f)(σ u L (i) ) z (q). u T \{} Since the coefficient vector of q is an eigenvector to the eigenvalue f(z), the sum over all u T \ {} has to be, D(u) z (f)(σ u L (i) ) z (q) =. u T \{} This holds for arbitrary polynomials f. Hence every (σ u L (i) ) z (q) has to vanish. In the given basis of, every operator is either a L (ji) or a proper shift. Therefore, every functional from the local dual basis annihilates q span N. This means q = and thus v = contradicting the assumption. Remark 3. Theorem 6 shows that the number of linearly independent common eigenvectors to the representation matrices is the number of irreducible primary components of the original ideal. This contrasts with the result of Theorem 3. There it was shown that the number of linearly independent common eigenvectors to the multiplication matrices is the number of maximal primary components, i.e. the cardinality of the variety. Hence the number of linearly independent common eigenvectors to the multiplication matrices is at most as large as the corresponding number for the representation matrices. As a byproduct to Theorem 6 we obtain a new method for computing radicals. Theorem 7. Let A P be a zero-dimensional ideal with variety V (A) = {z,..., z m } and normal set N = {t,..., t r }. Furthermore, let n v k E(i, B xi ), k =,..., m, and Then q := q + + q m satisfies i= q k := v T k t with t = (t,..., t r ) T. A : q = A.

14 526 H. M. Möller and R. Tenberg Proof. Let (k) be the Max Noether space of A at with basis {L (,k),..., L (s k,k) }, k =,..., m. By Theorems 4 and 5, every q k satisfies L (i,j) z j (q k ) = for i =,..., s j, j k, σ xl L (i,k) (q k ) = for i =,..., s k, l =,..., n. q k A because q k span N. Hence for an i = j(k) L (i,k) (q k ). Let f P. Then the Leibniz formula reduces to L (i,k) (f q) = f( )L (i,k) (q) = f( )L (i,k) (q k ). (9) If f A : q, then L (i,k) (f q) = for all i =,..., s k and k =,..., m. Using (9) with i = j(k) one gets f( ) =, k =,..., m, i.e. f A. If f A, then by (9) L (i,k) (f q) = for all i, k. This means fq A or f A : q. Example 3. We have in continuing the above example E(, B x ) E(, B y ) = span{( 2,, 2, ) T } and E(, B x ) E(, B y ) = span{(,,, ) T }. Choosing we get with t = (, x, y, x 2 ) T v := and v 2 := 2 q := 2 vk T t = y. k=. We compute a Gröbner basis of A : q similar to the FGLM technique as Lakshman pointed out in his thesis (Lakshman, 99). Because of NF(q) = q and 2 NF(xq) =, NF(yq) = 2 x2, NF(y 2 q) = 2 x2, we get x A : q, y 2 y A : q, A : q, and no y c in A : q, c a constant. Hence (x, y 2 y) = A : q = A. 7. How to Compute the Common Zeros Using Eigenvectors We already indicated in the preceding example how to compute an ideal quotient A : q if a normal form algorithm for A is known and if q is an arbitrary polynomial. The basic version of the algorithm is as follows (Lakshman, 99). Algorithm. (Ideal Quotient Computation) Given: A a zero-dimensional ideal, < an admissible ordering, and a procedure NF which computes for given f P the normal form of f w.r.t. an admissible ordering < 2. Output: A Gröbner basis G of A : q w.r.t. <.

15 Step. Initialize T o := T, the set of terms; G := ; N :=. Step 2. Let t o := min < T o ; T o := T o \ {t o }. Step 3. If NF(t o q) depends linearly on {NF(tq) t N }, NF(t o q) = t N Multivariate Polynomial System Solving 527 c t NF(tq), then enlarge G by t o t c tt and remove from T o all multiples of t o otherwise enlarge N by t o. Step 4. If T o then go to step 2, otherwise return G and stop. The FGLM-algorithm can be considered as special instance q =. For the practical computation of normal forms, recursive relations like NF(x i g) = NF(x i NF(g)) reduce the computational amount drastically. So one might think that the computation of the radical as given in Theorem 7 is not that difficult. The problem is to find an element v k of n i= E(i, B xi ). The computation of the sets n i= E(i, B xi ) or n i= E(i, M xi ) plays an essential role in the algorithm we propose in the following. It is based on the easily proved fact that v E(λ, A i ) = A j v E(λ, A i ) if A i A j = A j A i, and the fact that the matrices B x,..., B xn and M x,..., M xn resp. commute. We describe first the basic version of the algorithm, which holds without modification for both sets of matrices, B x,..., B xn and M x,..., M xn. Hence we use A i := M xi or A i := B xi resp. Algorithm. (Subspace Method, Basic Version) Given: Multiplication matrices (or representation matrices resp.) A,..., A n C r r. Output: The points of V (A). Initialization: Compute the set Λ of all eigenvalues of A and for every α Λ the eigenspace V α := E(α, A ). Loop: For j =,..., n compute for every V α,...,α j the set Λ α,...,α j of all eigenvalues α j+ of Φ α,...,α j : V α,...,α j V α,...,α j, v A j+ v. Denote by V α,...,α j+ the eigenspace corresponding to α j+. Final: Return all (α,..., α n ) corresponding to eigenspaces V α,...,α n. This algorithm terminates obviously. The result is correct because by commutativity of the matrices A,..., A n the mappings Φ α,...,α j are endomorphisms and by construction v V α,...,α j if and only if v is an eigenvector to eigenvalue α i of A i, i =,..., j. Hence n V α,...,α n = E(α i, A i ). i= n i= E(α i, A i ) is the null space if and only if (α,..., α n ) is not in V (A), but V α,...,α n is by construction not the null space. A shortcut to V α,...,α n is obtained, if dim V α,...,α j = for a j < n. Then all subsequent V α,...,α i, i > j, as non-null subspaces of V α,...,α j have dimension. Hence especially V α,...,α n = V α,...,α j. Therefore, an advanced version of the subspace method always tests the dimension of the actual V α,...,α j. If the dimension is, then V α,...,α n is found and the missing

16 528 H. M. Möller and R. Tenberg coordinates α j+,..., α n of the point α = (α,..., α n ) V (A) are either read off from D() α as described in Section 4 if A i = M xi or they are computed as eigenvalues of A i, i > j, if A i = B xi as described in Section 5. The advanced version works best, if the initial matrix A is already non-derogatory, i.e. if their eigenspaces all have dimension. If only an A j, j >, is non-derogatory, then V α,...,α j = j i= E(α i, A i ) has dimension and then the subsequent computation of eigenspaces is superfluous. We state explicitly that, in any case whenever a V α,...,α j is computed, there is at least one point of type (α,..., α j,...) in the variety. And if dim V α,...,α j >, then all subsequently computed eigenspaces usually have a smaller dimension. More precisely, dim V α,...,α j α j+ dim V α,...,α j+. The basic and the advanced version of the algorithm works with both types of matrices, the multiplication matrices M xi and the representation matrices B xi. But there are differences in the performance when applied to the two types. The subspace method using the representation matrices B xi has drawbacks compared with the method using the multiplication matrices M xi. First of all, the computation of eigenspaces needs floating point arithmetic. In the beginning, the Gröbner basis and hence the matrices B xi and M xi resp. are without rounding errors. But the successive computation of eigenspaces V α,...,α j, j =,..., n, induces rounding errors. The more computations are needed, the more the data are misrepresented. The dimension of the spaces V α,...,α j is an indicator of the complexity. In the case of multiplication matrices, the algorithm starts with an eigenspace V α of dimension at least m, where m denotes the number of distinct points of the variety with the same first component α. This means that here the multiplicity of the individual zeros is ignored, see Theorem 2. And at termination, the spaces V α,...,α n are all onedimensional by Theorem 3. This contrasts with the case of representation matrices. Here the spaces V α,...,α n can have a dimension greater than by Theorem 6 although every V α,...,α n corresponds to one single point of the variety. In a sense, the multiplicity of common zeros causes a greater dimension of eigenspaces in the B xi case. Only in the case of a complete intersection, i.e. A is generated by n polynomials f,..., f n, we know in advance that the usage of representation matrices also leads to one-dimensional common eigenspaces V α,...,α n (Eisenbud, 994, p. 529). A second drawback becomes apparent if the computation terminates with a onedimensional V α,...,α j, j < n. In the multiplication matrix case A i = M xi, the missing α k, k > j, can be read off without extra arithmetic operations apart from an eventual normalization. In the case A i = B xi however, the α k, k > j, have to be computed as eigenvalues of B xj given the eigenvector, the generator of V α,...,α j. This means for every k > j a matrix vector multiplication plus an eventual normalization. Example 4. We consider the ideal A given by the polynomials g (x, y, z) = y 3 y 2, g 2 (x, y, z) = z 2 4zx 25y 2 + 2x 8, g 3 (x, y, z) = zy + 3y 2 4y, g 4 (x, y, z) = x 2 6y 2 3x + 2,

17 g 5 (x, y, z) = xy 2y 2 2y. Multivariate Polynomial System Solving 529 The set G = {g, g 2, g 3, g 4, g 5 } is a Gröbner basis w.r.t. a degree ordering with y < x < z and we get N = {, y, x, z, y 2, zx}. The multiplication matrices w.r.t. x, y, and z are easily set up: M x :=, M y :=, M z := We first compute the eigenspaces of M x and obtain 4,, 2 as eigenvalues and with V 4 = span{v () 4 }, V = span{v (), v(2) }, V 2 = span{v () 2, v(2) 2, v(3) 2 } v () 4 = (,, 4,,, 4) T, v () = (,,,,, ) T, v (2) = (,,,,, ) T, v () 2 = (,, 2,,, ) T, v (2) 2 = (,,,,, 2) T, v (3) 2 = (,,,,, ) T. For α = 4 we have dim V 4 =, such that (,, 4,,, 4) T is a common eigenvector of all matrices and we read off the zero (4,, ) V (A). We have to apply M y on V and V 2 : and The eigenspaces of [ () M y v v (2) ] [ = () v v (2) [ () M y v 2 v (2) 2 v (3) ] [ 2 = () v 2 v (2) 2 v (3) 2 N = [ ] ] [ ] ] 4. and N 2 = 4

18 53 H. M. Möller and R. Tenberg are, as easy computation gives, Therefore in every case α 2 = and E(, N ) = span{(, ) T, (, ) T }, E(, N 2 ) = span{(,, ) T, (,, ) T }. V, = span{v (), v(2) }, V 2, = span{v () 2, v(2) 2 }. No V α,α 2 has dimension. Hence continuing with M z we get and [ () M z v v (2) ] [ = () v v (2) [ () M z v 2 v (2) ] [ 2 = () v 2 v (2) 2 The matrices [ ] 4 4 N = and N 2 = have only one eigenvalue each and eigenspaces E(2, N ) = span{(2, ) T }, E(4, N 2 ) = span{(, 4) T }. ] [ 4 4 ] ] [ ]. 6 8 [ ] 6 8 Therefore α 3 = 2 if (α, α 2 ) = (, ), α 3 = 4 if (α, α 2 ) = (2, ), and computation gives So we obtain three different roots: V,,2 = span{2v () + v (2) } = span{(,,, 2,, 2)T }, V 2,,4 = span{v () 2 + 4v (2) 2 } = span{(,, 2, 4,, 8)T }. z = (4,, ) from V 4 = V 4,,, z 2 = (,, 2) from V,,2, z 3 = (2,, 4) from V 2,,4. Of course, the computation of the eigenvectors generating V,,2 and V 2,,4 resp. was not necessary. It served only for illustrating Theorem 3. The computations for the representation matrices are similar. V 4 is spanned by v := (,,,,, ) T, the coefficient vector of y 2. The first component α is 4, the y- and z- components are eigenvalues of B y and B z resp. with corresponding eigenvector v each, resulting in α 2 = α 3 =. Computation gives, furthermore, V,,2 = span{(4,, 2, 2, 2, ) T }, V 2,,4 = span{( 4,, 4,, 9, ) T, (,,,,, ) T }. This implies that the Max Noether spaces at (4,, ) and (,, 2) are generated by one differential operator and its shifts, whereas the Max Noether space at (2,, 4) is generated by two differential operators.

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