Effective descent for differential operators

TL;DR

This paper develops an algorithmic method to find tensor decompositions of differential operators, building on Katz's theorem, and addresses the descent problem for explicit differential fields that are C1-fields.

Contribution

It introduces an algorithmic approach to compute the factors M and N in the tensor decomposition of differential operators, extending previous work on absolute irreducibility.

Findings

Provides an explicit algorithm for the descent problem in differential operators.

Applies to differential fields that are C1-fields, enabling practical computations.

Extends theoretical results of Katz with computational methods.

Abstract

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields.