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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 $$ σ(Y)=AY $$ where $A\in \mathrm{GL}_n(k(\mathbb{X})(x))$ and $σ$ is the shift operator $σ(x)=x+1$. 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 $S\subset k(\mathbb{X})[X,1/\det(X)]$. For a ${\bf c}\in \mathbb{X}$, denote by $v_{\bf c}$ the map from $k[\mathbb{X}]$ to $k$ given by $v_{\bf c}(f)=f({\bf c})$ for any $f\in k[\mathbb{X}]$. We prove that the set of ${\bf c}\in \mathbb{X}$ satisfying that $v_{\bf c}(A)$ and $v_{\bf c}(S)$ are well-defined and the affine variety in $\mathrm{GL}_n(k)$ defined by $v_{\bf c}(S)$ is the Galois group of $σ(Y)=v_{\bf c}(A)Y$ 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 $\mathrm{GL}_n(k)$ is the Galois group of a linear difference equation over $k(x)$ if and only if the quotient $G/G^\circ$ 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.