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4.1. THE GLOBAL MINIMUM OF A DOMINATED POLYNOMIAL 45
In certain applications it may be necessary to restrict the values of x 1 ,...,x n to,
for example, the positive orthant. However, in this research no attention is paid to
this form of constrained polynomial optimization. The extension of the techniques
described here to this more general setting is recommended for further research.
The global minimizers of p λ (x 1 ,...,x n ) are of course among the stationary points
of the polynomial p λ , which are the real solutions to the corresponding system of firstorder
conditions. This leads to a system of n polynomial equations in n variables of
the form:
d (i) (x 1 ,...,x n )=0, (i =1,...,n), (4.2)
where
d (i) (x 1 ,...,x n )=x 2d−1
i + 1 ∂
q(x 1 ,...,x n ). (4.3)
2dλ ∂x i
It will be convenient to write d (i) (x 1 ,...,x n ) in the form:
d (i) (x 1 ,...,x n )=x m i − f (i) (x 1 ,...,x n ), (i =1,...,n), (4.4)
with m =2d − 1 and f (i) = − 1 ∂
2dλ ∂x i
q(x 1 ,...,x n ) ∈ R[x 1 ,...,x n ] of total degree
strictly less than m. Because of the special structure of the polynomial system, (i)
the set of polynomials {d (i) | i =1,...,n} is in Gröbner basis form with respect to
any total degree monomial ordering and (ii) the associated variety V , the solution set
to the system of equations (4.2), has dimension zero which means that the system of
first-order conditions only admits a finite number of solutions in C n . The result that
there exists a finite number of solutions in this situation is based on Theorem 3.3 of
Section 3.2.4: for every i, 1≤ i ≤ n, there exists a power of x i which belongs to the
ideal 〈I〉, where I = {d 1 ,...,d n }. (For more details see Proposition 3.1 and Theorem
2.1 in [48] and the references given there.)
Note that because of property (i) it is not necessary to run an algorithm in order
to obtain a Gröbner basis for the ideal generated by the first-order conditions, because
the first-order conditions of a Minkowski dominated polynomial are in Gröbner
basis form already.
Because of these properties the Stetter-Möller matrix method, introduced in Section
3.3, can be applied to the system of equations (4.2). The associated ideal
I = 〈d (i) | i =1,...,n〉, yields the quotient space R[x 1 ,...,x n ]/I which is a finitedimensional
vector space of dimension N := m n . A monomial basis for this quotient
space is defined by the set:
B = {x α1
1 xα2 2 ···xαn n | α 1 ,α 2 ,...,α n ∈{0, 1,...,m− 1}}. (4.5)
For definiteness we will choose a permutation of the total degree reversed lexicographical
monomial ordering throughout this thesis, unless stated otherwise. For the
2-dimensional situation this ordering corresponds to:
1 ≺ x 1 ≺ x 2 1 ≺ ...≺ x 2d−2
1 ≺ x 2 ≺ x 1 x 2 ≺ x 2 1x 2 ≺ ...
(4.6)
≺ x 2d−2
1 x 2 ≺ x 2 2 ≺ ...≺ x 2d−2
1 x 2d−2
2