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8.3. A REPARAMETERIZATION OF THE MODEL REDUCTION PROBLEM 145
This yields the expression:
∏
k − δ l )
k>l,k,lj(δ ∏ k + δ i )
k>j(δ ∏ (−δ i − δ l )
j>l
m i,j =
∏
. (8.36)
(δ k − δ l )
k>l
Canceling common factors yields:
∏
k + δ i )
k>j(δ ∏ (−δ i − δ l )
j>l
m i,j = ∏
k − δ j )
k>j(δ ∏ (δ j − δ l )
j>l
(8.37)
= (−δ i − δ 1 ) ···(−δ i − δ j−1 )(−δ i − δ j+1 ) ···(−δ i − δ N )
.
(δ j − δ 1 ) ···(δ j − δ j−1 )(δ j − δ j+1 ) ···(δ j − δ N )
The denominator expression equals d ′ (δ j ), while the numerator equals − d(−δi)
(δ . The
i+δ j)
polynomial d(s) has its zeros at δ 1 ,δ 2 ,...,δ N in the open left-half plane Π − , which are
all of multiplicity 1. Therefore it holds that, d(−δ i ) 0, (δ i + δ j ) 0 and d ′ (δ j ) 0
for all pairs (i, j). Hence, the entries m i,j are non-zeros for all pairs (i, j), which
proves the theorem.
□
Note that Equation (8.32) can be regarded as a system of N equations in the
N unknowns x 1 ,...,x N with coefficients that depend on the parameters provided
by ρ(δ i ). For the case of co-order k = 1 the number of free parameters in ρ(δ i )is
k − 1 = 0. This case was studied in [50]. As in that paper, we intend to proceed here
by solving for the quantities x 1 ,x 2 ,...,x N , which then determine the polynomial
ã(s), from which q 0 and a(s) can be obtained. Subsequently, the polynomial b(s) can
be obtained as described in Section 8.6.
In the co-order k case, where k>1, the solutions x 1 ,x 2 ,...,x N should satisfy
the additional constraints ã N−1 = ··· =ã N−k+1 = 0 too. These constraints can in
view of Equation (8.27) be cast in the form of a linear constraint on the parameters
x 1 ,...,x N with coefficients that depend on δ 1 ,...,δ N only. Moreover, note that for
all values for k ≥ 1 a feasible solution requires that x i 0 for all i =1,...,N,so
that the trivial solution to the system of equations (8.32) must be discarded.
Furthermore it is important here to note that for k ≥ 1 the system of equations
(8.32) for the quantities x 1 ,x 2 ,...,x N is in Gröbner basis form with respect to any
total degree monomial ordering for fixed values of ρ 0 ,ρ 1 ,...,ρ k−1 . This observation is
crucial as it provides a way to solve this system of equations by applying the Stetter-
Möller matrix method mentioned in Section 3.3. This is studied in Section 8.5 of this
chapter.