# Explicit degree bounds for right factors of linear differential operators

## TL;DR

This paper provides explicit degree bounds for the right factors of linear differential operators with rational function coefficients, depending on the operator's degree, order, and local exponents, with applications over complex and number fields.

## Contribution

It introduces a fully explicit bound on the degrees of monic right factors of reducible linear differential operators, linking algebraic properties to local exponents and field characteristics.

## Key findings

- Explicit degree bounds depend on operator degree, order, and local exponents.
- Bounds are applicable over complex numbers and number fields.
- Results facilitate factorization analysis of differential operators.

## Abstract

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.