Border bases are a generalization of Gröbner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a string, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space.

翻译：边界基是多项式环中零维理想的 Gröbner 基的推广。在本文中，我们为线性微分算子的非交换环，即有理 Weyl 代数，引入了边界基。我们详细阐述了它们的性质，并提出了用于计算的算法。我们应用此理论将可积联络明确表示为循环 $D$-模。作为应用，我们探讨了弦理论、费曼积分以及宇宙学积分背后的微分方程。我们还讨论了具有固定纯量秩的特定 $D$-理想的分类，即常系数线性偏微分方程以及 Frobenius 理想的情形。我们的方法基于仿射空间中点的 Hilbert 概形理论。