特殊化下的差伽罗瓦群
Difference Galois groups under specialization
摘要：
本文给出了Hrushovski关于特殊化微分伽罗瓦群的一个结果的一个差分模拟.设$k$是特征为零的代数闭域，$\mathbb{X}$是$k$上的不可约仿射代数簇。考虑线性差分方程$$ σ(Y)=AY $$，其中$A\in \mathrm{GL}_n(k(\mathbb{X})(x))$和$σ$是移位算子$σ(x)=x+1$。假设在$\overline{k(\mathbb{X})}(x)$上的上述方程的伽罗瓦群$G$定义在$k(\mathbb{X})$上，即$G$的消失理想由有限集$S\subset k(\mathbb{X})[X,1/\det(X)]$生成。对于${\bf c}\in \mathbb{X}$，用$v_{\bf c}$表示由$v_{\bf c}(f)=f({\bf c})$为任何$f\in k[\mathbb{X}]$给出的从$k[\mathbb{X}]$到$k$的映射。证明了满足$v_{\bf c}(A)$和$v_{\bf c}(S)$是良定义的，且$v_{\bf c}(S)$定义的$\mathrm{GL}_n(k)$中的仿射簇是$k(x)$上的$σ(Y)=v_{\bf c}(A)Y$的Galois群的${\bf c}\in \mathbb{X}$的集合在$\mathbb{X}$中是Zerki稠密的.我们应用我们的结果范德普特-辛格的猜想断言，一个代数子群$G$的$\mathrm{GL}_n(k)$是伽罗瓦群的线性差分方程在PDMATHANF当且仅当商PDMATHANG的单位分量是循环的。我们表明，如果范德普-辛格猜想是真的$k=\mathbb{C}$，那么它将是真的任何代数闭域$k$的特征为零。
英文摘要：
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.
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