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36 CHAPTER 3. SOLVING POLYNOMIAL SYSTEMS OF EQUATIONS
interpreted as the x i -values of the zeros of the ideal I. A point (λ 1 ,...,λ n ) ∈ C n
of eigenvalues of A T x 1
,...,A T x n
, respectively, corresponding to a common (non-zero)
eigenvector v ∈ C n is also called a multi-eigenvalue of the n-tuple of commuting
matrices (A T x 1
,...,A T x n
).
The following theorem characterizes the variety V (I):
Theorem 3.4. V (I) ={(λ 1 ,...,λ n ) ∈ C n : ∃v ∈ C n \{0} ∀i : A T x i · v = λ i · v}.
Proof. See [48] for the proof of this theorem.
□
The eigenvectors v of the matrices A T x 1
,...,A T x n
are highly structured. If all
the eigenvalues of these matrices have algebraic multiplicity one, then it is wellknown
that A T x i
= UΛL T , where L T U = I and U is a generalized Vandermonde
matrix. The rows of U T consist of the ordered basis B evaluated at the points
x = v(k) ∈ C n , k =1,...,N, where v(1),v(2),...,v(N) are the complex solutions
to the system of equations (3.19). Furthermore Λ = diag(λ 1 ,...,λ N ) is a diagonal
matrix with λ k = r(v(k)), k =1, 2,...,N. The matrix L has the property that its
k-th column consists of the coefficients l jk of the Lagrange interpolation polynomials
l k (x 1 ,...,x n )= ∑ N
j=1 l j,kb(j), where b(j) denotes the j-th basis element in B, with
the basic interpolation property that l k (v(k)) = 1 and l k (v(j)) = 0 for all j k. In
linear algebra terms this says that L T = U −1 . The fact that the eigenvectors of A T x i
are columns of the generalized Vandermonde matrix U was noted by Stetter, which
is the reason that eigenvectors of this form are sometimes called Stetter vectors [81].
The structure of these vectors can be used to read off the point v(k) from an eigenvector
of a single matrix A T x i
directly, without having to apply the eigenvectors to all
the operators A T x 1
,...,A T x n
, to obtain the corresponding eigenvalues. This is because
the monomials x i , i =1,...,n, are all in the basis B, hence their values appear in
the Stetter vector (see also [20]).
The method of rewriting the problem of finding solutions of a set of polynomial
equations into an eigenvalue problem of a set of commuting matrices, is called the
Stetter-Möller matrix method and can only be applied to systems of polynomial equations
which generate a zero-dimensional ideal. This method is described in [81] and
a similar approach can be found in [50] and [48] in which the matrix method was
derived independently of the work of Stetter and Möller. For a further background
on the constructive algebra and systems theory aspects of this approach, see also [47].
3.4 Counting real solutions
For some specific applications only real solutions are of importance. Therefore it can
be convenient to have information about the number of real and complex solutions
of an ideal I generated by polynomials in R[x 1 ,...,x n ] of a system of equations in
Gröbner basis form. The numbers of real and complex solutions can be computed by