On a classification of polynomial differential operators with respect to the type of first integrals.

This paper gives a classification of polynomial differential operators X = X 1 ( x 1 , x 2 ) δ 1 + X 2 ( x 1 , x 2 ) δ 2 ( δ i = ∂ / ∂ x i ) . The classification is defined through an order derived from X . Let X = X y be the associated differential polynomial, the order is defined as the order of a differential ideal Λ that is an essential extension of { X } . The main result shows the order can only be four possible values: 0, 1, 2, 3, or ∞. Furthermore, when the order is finite, the essential extension Λ = { X , A } , where A is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of X . When the order is infinite, the extension Λ is identical with { X } . In addition, if, and only if, the order is 0, 1, or 2, the associated polynomial differential equation has Liouvillian first integrals. Examples and connections with Godbillon–Vey sequences are also discussed. [ABSTRACT FROM AUTHOR]