Difference Galois groups under specialization

Abstract:

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 let $\mathbb {X}$ be an irreducible affine algebraic variety over $k$. Consider the linear difference equation \begin{equation*} \sigma (Y)=AY, \end{equation*} where $A\in \mathrm {GL}_n(k(\mathbb {X})(x))$ 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 $S\subset k(\mathbb {X})[X,1/\det (X)]$. For a ${\mathbf {c}}\in \mathbb {X}$, denote by $v_{{\mathbf {c}}}$ the map from $k[\mathbb {X}]$ to $k$ given by $v_{{\mathbf {c}}}(f)=f({\mathbf {c}})$ for any $f\in k[\mathbb {X}]$. We prove that the set of ${\mathbf {c}}\in \mathbb {X}$ satisfying that $v_{\mathbf {c}}(A)$ and $v_{\mathbf {c}}(S)$ are well-defined and the affine variety in $\mathrm {GL}_n(k)$ 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 $\mathrm {GL}_n(k)$ 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.