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54 CHAPTER 5. ND-SYSTEMS APPROACH IN POLYNOMIAL OPTIMIZATION
where the nD-shift operates on such state vectors in an element-wise fashion. Since
this operator is linear, the latter relation can also be cast in the usual matrix-vector
form. This requires a choice of basis. If this choice is made to correspond to the basis
B for the quotient space R[x 1 ,...,x n ]/I and the associated monomial ordering, it
holds that:
w t1,...,t i−1, t i+1, t i+1,...,t n
= σ i w t1,t 2,...,t n
= A T x i
w t1,t 2,...,t n
, (5.7)
where the matrix A T x i
again denotes the matrix associated with multiplication by the
polynomial x i within the quotient space R[x 1 ,...,x n ]/I. Note that its transpose A T x i
is involved in this relationship. As a consequence the following holds:
w t1+α 1,t 2+α 2,...,t n+α n
=(A T x 1
) α1 (A T x 2
) α2 ···(A T x n
) αn w t1,t 2,...,t n
, (5.8)
which shows that the general solution to the autonomous multidimensional system
with initial state w 0,0,...,0 is given by:
w t1,t 2,...,t n
=(A T x 1
) t1 (A T x 2
) t2 ···(A T x n
) tn w 0,0,...,0 . (5.9)
More generally, for an arbitrary polynomial r(x 1 ,...,x n ) it holds that:
r(σ 1 ,...,σ n ) w t1,...,t n
= A T r(x 1,...,x n) w t 1,...,t n
. (5.10)
Note that y t1,t 2,...,t n
can be read off from this as being an element of w t1,t 2,...,t n
.
Now we intend to use the recursions in Equation (5.3) rather than Equation (5.9)
to compute w t1,t 2,...,t n
(and y t1,t 2,...,t n
). In this way the action of a matrix A T r(x 1,...,x n)
on a given vector v can be computed without constructing this matrix explicitly: only
the relations between the time series as represented by (5.3) are required. The values
of the time series in the initial state w 0,...,0 are known as they are chosen as the values
in the given vector v.
A more detailed formulation of the nD-systems approach is described in Theorem
3.1 of [13].
Recall that our interest is in computing the eigenvalues of the matrix
A r(x1,...,x n), which coincide with the eigenvalues of its transpose A T r(x 1,...,x n)
and which
may conveniently be studied from the perspective of the autonomous nD-system
introduced above. Note that if v is an eigenvector of A T x i
with a corresponding
eigenvalue ξ i , then it holds that:
A T x i
v = ξ i v. (5.11)
In terms of the nD-system this implies that the choice w 0,...,0 := v for the initial
state produces a scaled version for the state: w 0,...,0,1,0,...,0 = ξ i v, which relates to a
shift in the multidimensional time space by 1 in the direction of the i-th time axis
only. However, the vectors w 0,...,0 and w 0,...,0,1,0,...,0 have m n − m (n−1) elements in
common (in shifted positions), showing that the eigenvectors of A T x i
exhibit a special
structure. This corresponds to the Stetter vector structure discussed in Chapter 4.