🤖 AI Summary
Boundary basis theory—previously developed for commutative polynomial rings—lacked a noncommutative extension, particularly for the rational Weyl algebra (the ring of linear differential operators), hindering Gröbner-free computational approaches to D-modules.
Method: Integrating D-module theory, Hilbert schemes of points, and computational algebraic geometry, we develop a Gröbner-free framework for border bases over the rational Weyl algebra, tailored to integrable connections and cyclic D-modules.
Contributions/Results: (1) We provide explicit D-module representations for integrable connections; (2) we achieve a systematic classification of D-ideals of fixed holonomic rank (equivalently, homological rank), encompassing constant-coefficient linear PDE systems and Frobenius ideals; (3) we explicitly determine solution structures for key differential systems arising in string theory, Feynman integral computations, and cosmology. This work establishes the first theory of noncommutative border bases, filling a foundational gap and introducing a new paradigm for physics-driven D-module computation.
📝 Abstract
Border bases are a generalization of Gröbner bases for 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 stringy, 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.
