Submodule approach to creative telescoping
Mark van Hoeij
TL;DR
This work reframes creative telescoping as annihilator computation in $D$-modules and introduces a submodule perspective that yields factorized, faster telescopers. By selecting a nontrivial submodule $N\subsetneq M$, the authors obtain a right-factor $R$ of the full telescoper $L$ via $R = {\rm Ann}^{\min}_D(m, M/N)$ and a left factor $L' = {\rm Ann}^{\min}_D( R(m), N)$, enabling a cyclic-vector computation to determine $L'$. Exploiting automorphisms of $N$ further decomposes the problem into component telescopers whose LCLMs produce compact, interpretable factored forms, demonstrated on hypergeometric terms with substantial speedups over standard expansions. The approach offers both practical computational benefits and theoretical clarity on why minimal telescopers may dominate minimal recurrences, and it raises open questions about automatic submodule discovery and invariant data extraction from $D$-modules.
Abstract
This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator $L \in D$ for an element $m$ in a $D$-module $M$. The main idea is to look for submodules of $M$. If $N$ is a non-trivial submodule of $M$, constructing the minimal operator $R$ of the image of $m$ in $M/N$ gives a right-factor of $L$ in $D$. Then $L = L' R$ where the left-factor $L'$ is the telescoper of $R(m) \in N$. To expedite computing $L'$, compute the action of $D$ on a natural basis of $N$, then obtain $L'$ with a cyclic vector computation. The next main idea is that when $N$ has automorphisms, use them to construct submodules. An automorphism with distinct eigenvalues can be used to decompose $N$ as a direct sum $N_1 \oplus \cdots \oplus N_k$. Then $L'$ is the LCLM (Least Common Left Multiple) of $L_1, \ldots, L_k$ where $L_i$ is the telescoper of the projection of $R(m)$ on $N_i$. An LCLM can greatly increase the degrees of coefficients, so $L'$ and $L$ can be much larger expressions than the factors $L_1,\ldots,L_k$ and $R$. Examples show that computing each factor $L_i$ and $R$ seperately can save a lot of CPU time compared to computing $L$ in expanded form with standard creative telescoping.