Explicit degree bounds for right factors of linear differential operators

Alin Bostan (1) , Tanguy Rivoal (2) , Bruno Salvy (3)

1 SPECFUN - Symbolic Special Functions : Fast and Certified
2 IF - Institut Fourier
3 ARIC - Arithmetic and Computing

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominators of 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.

Mots clés: Factorization; Fuchs’ relation; Differential operators

Domaines: Analyse classique [math.CA]; Théorie des nombres [math.NT]

Alin Bostan, Tanguy Rivoal, Bruno Salvy. Explicit degree bounds for right factors of linear differential operators. 2019. ⟨hal-02154679v3⟩