Description
Abstract.This talk 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 $H$ in a D-module $M$. The main idea is to look for submodules of $M$. For a non-trivial submodule $N$, constructing the minimal operator $R$ of the image of $H$ in $M/N$ gives a right-factor of $L$ in $D$. Then $L = L' R$ where $L'$ is the telescoper of $R(H)$. To expedite computing $L'$, compute the action of $D$ on a natural basis of $N$, then obtain the telescoper $L'$ for $R(H)$ 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 of submodules $N_1, \ldots, N_k$. Then $L' = \text{LCLM}(L_1, \ldots, L_k)$ where $L_i$ is the telescoper of the projection of $R(H)$ on $N_i$. An LCLM can greatly increase the degrees of the coefficients, so $L'$ and hence $L$ can be much larger than the factors $L_1, \ldots, L_k$ and $R$. Examples show that computing each factor $L_i$ and $R$ separately can save a lot of CPU time compared to computing the full telescoper $L$ all at once with standard creative telescoping.
