Effective Computation of Centralizers of ODOs

Abstract This work is devoted to the symbolic computation of centralizers of ordinary differential operators (ODOs), in the ring of differential operators. Starting with an operator $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> of order $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> and the order $\mathfrak{m}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> of a non-trivial operator in its centralizer, which is not a multiple of $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> , a finite set of generators of a subalgebra of the centralizer is obtained, maximal of a certain rank $R$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , the greatest common divisor of all orders of its elements. The true rank $r$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> of the centralizer is unknown to start unless $(n,\mathfrak{m})=1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> since $1\leq r\leq R\leq (n,\mathfrak{m})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>r</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>R</mml:mi> <mml:mo>≤</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:math> , and remains unknown unless our algorithm returns $R=1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> . Ours is a direct approach based on solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchies, which after substituting the coefficients of $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> belong to a computable differential field. In addition, by considering parametric coefficients, we develop an algorithm to generate families of ODOs with non-trivial centralizer, whose coefficients belong to a previously chosen differential field. Our algorithms are implemented in SageMath.