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3
with
for some
u
∈
R
u
= 1. Also, we consider the worst-case translational error
p
∗
∞
≤
ε
.
R
,
t
−
s
t
(
ε
)=sup
t
s.t.
p
(9.9)
In the sequel we will consider without loss of generality that
F
∗
coincides with
F
abs
.
9.2.3
SMR of Polynomials
The approach proposed in this chapter is based on the SMR of polynomials intro-
duced in [13] to solve optimization problems over polynomials via LMI techniques.
Specifically, let
y
(
x
) be a polynomial of degree 2
m
in the variable
x
=(
x
1
,...,
x
n
)
T
∈
n
,
i.e.
R
∑
y
i
1
,...,
i
n
x
i
1
···
x
i
n
y
(
x
)=
i
1
+
...
+
i
n
≤
2
m
i
1
≥
0
,...,
i
n
≥
0
for some coefficients
y
i
1
,...,
i
n
∈
R
. According to the SMR,
y
(
x
) can be expressed as
y
(
x
)=
v
(
x
)
T
(
Y
+
L
(
α
))
v
(
x
)
where
v
(
x
) is any vector containing a base for the polynomials of degree
m
in
x
,and
hence can be simply chosen such that each of its entry is a monomial of degree less
than or equal to
m
in
x
, for example
x
1
,
x
n
)
T
v
(
x
)=(1
,
x
1
,...,
x
n
,
x
1
x
2
,...,
.
The matrix
Y
is any symmetric matrix such that
y
(
x
)=
v
(
x
)
T
Yv
(
x
)
which can be simply obtained via trivial coefficient comparisons. The vector
α
is a
vector of free parameters, and the matrix function
L
(
α
) is a linear parametrization
of the linear set
L
=
L
T
:
v
(
x
)
T
Lv
(
x
)=0
L
=
{
∀
x
}
which can be computed through standard linear algebra techniques for parameteriz-
ing linear spaces.
The matrices
Y
and
Y
+
L
(
) are known as SMR matrix and complete SMR
matrix of
y
(
x
). The length of
v
(
x
) is given by
α
m
)=
(
n
+
m
)!
n
!
m
!
d
1
(
n
,
while the length of
α
(
i.e.
, the dimension of
L
)is
m
)=
1
d
2
(
n
,
2
d
1
(
n
,
m
)(
d
1
(
n
,
m
)+1)
−
d
1
(
n
,
2
m
)
.
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