By M. Petrich

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E x e r c i s e 4 . 5). T h a t is: (1) find an algorithm to construct all rankings, (2) find an algorithm to decide if two constructed rankings are equal (and if possible, find a canonical representation of a ranking). E x e r c i s e 4 . 7 (Difficult) Characterize the set of all rankings for t h e general case, t h a t is, for m derivation operators and n differential indeterminates. We now assume t h a t a ranking is fixed and given on QY. To avoid verbal description of certain polynomial subrings of 31 = &{ yi, • • •, yn }, it is convenient to a d a p t the suggestive notation of Morrison [29].

7 In Ritt [32] or Kolchin [23], the "rank" of a differential polynomial is not explicitly defined and is only used in the comparative sense. If yj is the highest ranked variable in an algebraic polynomial F and if d is the degree of F in yj, the index j is sometimes called the class of F and the degree d is called the class degree of / . These terms are commonly used in the literature on geometric theorem proving based on the Ritt-Wu method. 13 Let m = 1 and let the ranking be induced by the lexicographical order with respect to the tuple (j,h), where u = Shyj G QY.

To prove the converse, note that I\ is by definition invertible with respect to the empty set. Suppose by induction, we have proved that for some 2 < k < p, Ij is invertible with respect to Aj_i for j < k. Let G G Sfc_i • ((A fe _i):/£°). 11). , Gi. By hypothesis, G\ = 0. 16 applied to Afc_i, Ik is invertible with respect to Afc_i. 6 also holds. 18). 22 Suppose A is triangular as before. (a) The property that A has invertible initials is decidable. (b) If A has invertible initials, and F G S p , then the invertibility of F with respect to A is decidable.