link to my thesis

5.1. THE ND-SYSTEM 53
cerned with the action of A T r rather than with the action of A r , as this will allow for
a meaningful system-theoretic interpretation.
To this end we set up in discrete time an autonomous multidimensional system,
also called an nD-system, associated with the set of polynomials d (i) ,(i =1,...,n)
representing the first-order conditions as in (4.2). With any monomial x α1
1 xα2 2 ···xαn n
we associate an nD-shift operator σ α1
1 σα2 2 ···σαn n which acts on any multidimensional
time series y t1,t 2,...,t n
according to the rule:
σ α1
1 σα2 2 ···σαn n : y t1,t 2,...,t n
↦→ y t1+α 1,t 2+α 2,...,t n+α n
. (5.1)
Imposing the usual linearity properties, this allows one to associate a homogeneous
multidimensional difference equation with an arbitrary polynomial r(x 1 ,x 2 ,...,x n )
as follows:
r(σ 1 ,σ 2 ,...,σ n ) y t1,t 2,...,t n
=0. (5.2)
Applying this set-up to the system of polynomial equations d (i) for i =1,..., n
as in (4.2) a system of n linear homogeneous multidimensional difference equations is
obtained, which can be written in the form:
y t1,...,t i−1, t i+m, t i+1,...,t n
= f (i) (σ 1 ,...,σ n ) y t1,t 2,...,t n
, (i =1,...,n) (5.3)
∂
where f (i) = − 1
2dλ ∂x i
q(x 1 ,...,x n ) ∈ R[x 1 ,...,x n ] of total degree strictly less than
m (in the same way as in Equation (4.4)).
Let for each multidimensional time instant t =(t 1 ,...,t n ) the ‘total time’ be
denoted by |t| := t 1 + ... + t n . Equation (5.3) expresses the fact that the value
of y¯t 1,¯t 2,...,¯t n
at any multidimensional ‘time instant’ ¯t = (¯t 1 , ¯t 2 ,...,¯t n ), such that
max{¯t 1 , ¯t 2 ,...,¯t n } is greater than or equal to m, can be obtained from the set of values
of y t1,t 2,...,t n
for which the multidimensional time instants have a total time |t| := t 1 +
t 2 +...+t n strictly less than the total time |¯t | = ¯t 1 +¯t 2 +...+¯t n . As a consequence, any
multidimensional time series y t1,t 2,...,t n
satisfying this system of recursions is uniquely
determined by the finite set of (total degree reversed lexicographically ordered) values:
w 0,0,...,0 := {y t1,t 2,...,t n
| t 1 ,t 2 ,...,t n ∈{0, 1,...,m− 1}}. (5.4)
Conversely, each choice for w 0,0,...,0 yields a corresponding solution for y t1,t 2,...,t n
.
In state-space terms, the set of values w 0,0,...,0 acts as an initial state for the autonomous
homogeneous system of multidimensional difference equations (5.3). This
point of view can be formalized by introducing the state vector w t1,t 2,...,t n
at the
multidimensional time instant (t 1 ,t 2 ,...,t n ) as the set of values:
w t1,t 2,...,t n
:= {y t1+s 1,t 2+s 2,...,t n+s n
| s 1 ,s 2 ,...,s n ∈{0, 1,...,m− 1}}. (5.5)
The elements of the state vector w t1,t 2,...,t n
should be ordered again by the total
degree reversed lexicographical ordering.
According to this definition, two state vectors w t1,t 2,...,t n
and w t1+α 1,t 2+α 2,...,t n+α n
,
with α i ≥ 0 for all i =1,...,n, are related by:
w t1+α 1,t 2+α 2,...,t n+α n
= σ α1
1 σα2 2 ···σαn n w t1,t 2,...,t n
, (5.6)