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 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.