We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.

中文翻译：

---

代数群作为线性微分方程的差分伽罗瓦群

我们研究了差分-微分域上线性微分方程的差分伽罗瓦理论中的反问题$\mathbb{C}(X)$ 与推导 $\frac{d}{dX}$ 和内同态 $F(X)\mapsto F(X+1)$. 我们的主要结果是每个线性代数群，被认为是一个差分代数群，作为一些线性微分方程的差分伽罗瓦群出现$\mathbb{C}(X)$.