We present a difference analogue of a result given by Hrushovski on differential Galois groups under specialization. Let
k
k
be an algebraically closed field of characteristic zero and let
X
\mathbb {X}
be an irreducible affine algebraic variety over
k
k
. Consider the linear difference equation
σ
(
Y
)
=
A
Y
,
\begin{equation*} \sigma (Y)=AY, \end{equation*}
where
A
∈
G
L
n
(
k
(
X
)
(
x
)
)
A\in \mathrm {GL}_n(k(\mathbb {X})(x))
and
σ
\sigma
is the shift operator
σ
(
x
)
=
x
+
1
\sigma (x)=x+1
. Assume that the Galois group
G
G
of the above equation over
k
(
X
)
¯
(
x
)
\overline {k(\mathbb {X})}(x)
is defined over
k
(
X
)
k(\mathbb {X})
, i.e., the vanishing ideal of
G
G
is generated by a finite set
S
⊂
k
(
X
)
[
X
,
1
/
det
(
X
)
]
S\subset k(\mathbb {X})[X,1/\det (X)]
. For a
c
∈
X
{\mathbf {c}}\in \mathbb {X}
, denote by
v
c
v_{{\mathbf {c}}}
the map from
k
[
X
]
k[\mathbb {X}]
to
k
k
given by
v
c
(
f
)
=
f
(
c
)
v_{{\mathbf {c}}}(f)=f({\mathbf {c}})
for any
f
∈
k
[
X
]
f\in k[\mathbb {X}]
. We prove that the set of
c
∈
X
{\mathbf {c}}\in \mathbb {X}
satisfying that
v
c
(
A
)
v_{\mathbf {c}}(A)
and
v
c
(
S
)
v_{\mathbf {c}}(S)
are well-defined and the affine variety in
G
L
n
(
k
)
\mathrm {GL}_n(k)
defined by
v
c
(
S
)
v_{{\mathbf {c}}}(S)
is the Galois group of
σ
(
Y
)
=
v
c
(
A
)
Y
\sigma (Y)=v_{{\mathbf {c}}}(A)Y
over
k
(
x
)
k(x)
is Zariski dense in
X
\mathbb {X}
.
We apply our result to van der Put-Singer’s conjecture which asserts that an algebraic subgroup
G
G
of
G
L
n
(
k
)
\mathrm {GL}_n(k)
is the Galois group of a linear difference equation over
k
(
x
)
k(x)
if and only if the quotient
G
/
G
∘
G/G^\circ
by the identity component is cyclic. We show that if van der Put-Singer’s conjecture is true for
k
=
C
k=\mathbb {C}
, then it will be true for any algebraically closed field
k
k
of characteristic zero.