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10.1. SOLVING THE SYSTEM OF QUADRATIC EQUATIONS 165
The values of ρ 1 that make the polynomial matrix Ãã N−1
(ρ 1 ) T singular are called
the polynomial eigenvalues of Ãã N−1
(ρ 1 ) T . Thus, the problem is transformed into the
following polynomial eigenvalue problem:
ÃãN−1 (ρ 1 ) T v = 0 (10.10)
where ρ 1 is the unknown parameter and v the eigenvector.
As a corollary on the degree of the parameter ρ 1 in the involved matrix we have:
Corollary 10.2. The polynomial degree of the matrix Ãã N−1
(ρ 1 ) T with respect to ρ 1
is equal to N − 1.
Proof. When carrying out multiplication of a row of AãN−1 (ρ 1 ) T by the associated
polynomial factor ρ(δ 1 ) α1 ρ(δ 2 ) α2 ···ρ(δ N ) α N
, the polynomial degree may increase to
at most N. The latter may potentially only occur when all the α i are equal to 1; i.e.,
when addressing the row of AãN−1 (ρ 1 ) T containing the coefficients of the normal form
of ã N−1 (x 1 ,x 2 ,...,x N ) x 1 x 2 ···x N . However, in that case all the N terms initially
need reduction, so that all the terms have at least one factor ρ(δ i ) in their denominator.
Multiplication by ρ(δ 1 )ρ(δ 2 ) ···ρ(δ N ) therefore leads to some cancelation of
factors for each term in the normal form, so that the resulting polynomial degree is
at most N − 1.
It is easily verified that the degree N − 1 does in fact occur for the polynomial
coefficient of x 1 x 2 · x N in the normal form of each of the quantities ρ(δ 1 ) ···ρ(δ i−1 )
ρ(δ i+1 ) ···ρ(δ N ) ã N−1 (x 1 ,x 2 ,...,x N ) x 1 ··· x i−1 x i+1 ···x N . To see this, note that
no reduction step for each term needs to be considered, because the total degree drops
below N as soon as a reduction step is applied. Therefore, the coefficient of x 1 x 2 · x N
in the normal form equals ρ(δ 1 ) ···ρ(δ i−1 )ρ(δ i+1 ) ···ρ(δ N )γ i,i . Since γ 1,i = 1
d ′ (δ i) 0
and since the expressions ρ(δ j )=1+δ j ρ 1 are of degree 1 (δ j is in Π − , hence non-zero),
the degree N − 1 is indeed achieved.
□
To summarize: to find all the solutions of the system of equations (10.2), one needs
to compute: (i) all the eigenvalues ρ 1 such that the polynomial matrix Ãã N−1
(ρ 1 ) T becomes
singular, and (ii) all the corresponding eigenvectors v, for which Ãã N−1
(ρ 1 ) T v =
0, from which to read off the values of x 1 ,...,x N . The N-tuples of values (x 1 ,...,x N ),
thus obtained, together with the corresponding polynomial eigenvalues ρ 1 , constitute
all the solutions of the quadratic system of equations (10.2) which satisfy the constraint
ã N−1 (x 1 ,..., x N ) = 0 in (10.3).
In the next section we show an approach to solve the polynomial eigenvalue problem
Ãã N−1
(ρ 1 ) T v = 0 in (10.10) by linearizing it into a generalized eigenvalue problem
in order to obtain the eigenvalues ρ 1 and the eigenvectors v.