The Groebner fan of a differential ideal in the ring of differential polynomials

By the Newton's lemma the author mean that the coefficient of the dominant term of a series solution of an equation satisfies the characteristic equation related to an edge of the Newton polygon of the equation, while the dominant exponent equals the slope of the edge. Originally, Newton considered an algebraic equation, later these considerations have been extended to systems of equations and also to differential equations. The Newton's lemma is proved in a general setting of the valuations on extensions of the ring of Laurent polynomials , as solutions, and on the ring of differential polynomials, induced by linear functions on exponent vectors. This allows the author to introduce the Groebner fan of a differential ideal.