Difference Galois groups under specialization

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