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

Let L=@?0@?k@?n(f"k/f)d^kd^kx be a linear differential operator with rational coefficients, where f"k,f@?@?@?[X] and @?@? 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 @?)^0^(^n^^^4^) where @? is the bit size of L. Moreover, a bound N @? exp((@?2^n)^2^n) is obtained. We also exhibit a polynomial time algorithm for calculating the greatest common (right) divisor of a family of operators.

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