Border Bases in the Rational Weyl Algebra

TL;DR

This paper extends the concept of border bases to the non-commutative rational Weyl algebra, providing algorithms and applications for representing integrable connections and classifying specific D-ideals.

Contribution

It introduces border bases for the rational Weyl algebra, a non-commutative setting, and develops algorithms for their computation and applications in differential equations.

Findings

Border bases are successfully generalized to the rational Weyl algebra.

Algorithms for computing border bases in this non-commutative setting are presented.

Applications include explicit representations of integrable connections and classification of D-ideals.

Abstract

Border bases are a generalization of Gr\"obner 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.