# Complexity of Janet basis of a D-module

## TL;DR

This paper establishes a double-exponential upper bound on the degree and complexity of Janet bases for D-modules, extending known bounds from polynomial modules to a more general non-commutative setting.

## Contribution

It provides the first known double-exponential bound for Janet bases of D-modules, highlighting a significant complexity result in non-commutative algebra.

## Key findings

- Double-exponential upper bound on degree and complexity
- Generalization from polynomial modules to D-modules
- Bound cannot be derived directly from commutative case

## Abstract

We prove a double-exponential upper bound on the degree and on the complexity of constructing a Janet basis of a $D$-module. This generalizes a well known bound on the complexity of a Gr\"obner basis of a module over the algebra of polynomials. We would like to emphasize that the obtained bound can not be immediately deduced from the commutative case.