A Shape Lemma for Ideals of Differential Operators

TL;DR

This work extends the classical shape lemma from commutative zero-dimensional ideals to zero-dimensional left ideals in the noncommutative algebra of differential operators, enabling a structured representation of annihilating ideals for D-finite functions. It introduces the notions of normal position and D-radical, proving a differential Shape Lemma that characterizes when a zero-dimensional differential-operator ideal can be generated by a single operator in $K[D_x]$ and $n$ shift terms $D_{y_i}-Q_i$, with $Q_i\in K[D_x]$ of lower order than a defining $P$. A key contribution is showing that every D-radical ideal can be transformed to normal position by a linear change of variables $\boldsymbol{y}\leftarrow\boldsymbol{y}+\boldsymbol{c}x$, aligning with the commutative analogue and facilitating constructive elimination for integrals via creative telescoping. The results clarify when a gauge-transform approach is needed versus when a change of variables suffices, and they set the stage for extending these ideas to recurrence operators, though the latter remains open.

Abstract

We propose a version of the classical shape lemma for zero-dimensional ideals of a commutative multivariate polynomial ring to the noncommutative setting of zero-dimensional ideals in an algebra of differential operators.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Definition 6
• Example 7
• Theorem 8
• proof