Complexity of factoring and calculating the GCD of linear ordinary differential operators

Abstract

Let L = ∑ 0 ≤ k ≤ n ( f k / f ) d k d k x be a linear differential operator with rational coefficients, where f k , f ∈ ℚ ¯ [ X ] and ℚ ¯ is the field of algebraic numbers. Let deg ⁡ x ( L ) = max ⁡ 0 ≤ k ≤ n { deg ⁡ x ( f k ) , deg ⁡ x ( f ) } and let N be an upper bound on deg x ( L j ) for all possible factorizations of the form L = L 1 L 2 L 3 , where the operators L j are of the same kind as L and L 2, L 3 , are normalized to have leading coefficient 1. An algorithm is described that factors L within time (N ℒ) 0 (n 4) where ℒ is the bit size of L. Moreover, a bound N ≤ exp((ℒ2 n ) 2 n ) is obtained. We also exhibit a polynomial time algorithm for calculating the greatest common (right) divisor of a family of operators.