[For the entire collection seeZbl 0606.00020.]
The present paper is a short exposition of J. F. Ritt’s theory of differential algebra, simplified to give only an overview and modified to emphasize the constructive character of that theory. The main ideas consist of a strong algebraic formalization of the methods (usually intuitively applicated) which lead to the solution of a system of differential equations in which the unknown functions and their derivatives as well as the coefficient functions occur in polynomial form. In addition to the ordinary polynomial algebra, differentiation is to be thought of as a further algebraic operation. Similar to the theory of Gröbner bases, a lexicographical ordering of the polynomials and the concepts of initial(P) and separant(P) of a polynomial P are introduced (let \(Y_ p\) be the highest-order variable and \(D^ mY_ p\) its highest derivative occurring in P with the highest power d, then initial(P) is the coefficient at \((D^ mY_ p)^ d\) and \(separant(P)=\partial P/\partial D^ mY_ p)\). With these concepts an ascending set (S) (playing the role of an ideal base) and a remainder of a polynomial with respect to (S) are defined. The main theorem (Ritt’s principle) consists of an algorithm to decide in a finite number of steps whether a given system (DP) is contradictory or not, and - in the latter case - of constructing an ascending set (CS) for it (such that each \(P\in (DP)\) has remainder 0 with respect to (CS)).
The theory is then refined in order to include conditions of nondegeneracy, since most theorems are not universally true but only under such additional conditions. Furthermore the paper reports that these algorithms were implemented on some small computers and several theorems of differential geometry could be proved in this way. As an example, the theory of a pair of Bertrand curves is elaborated in more detail.

Reviewer: W.Degen