It is common knowledge that algebra, including algebraic geometry, historically grew out of the study of algebraic equations with numerical coefficients. In much the same way, differential algebra sprang from the classical study of algebraic differential equations with coefficients that are meromorphic functions in a region of some complex space C". As a consequence, differential algebra bears a considerable resemblance to the elementary parts of algebraic geometry. Indeed, since an algebraic equation can be considered as a differential equation in which derivatives do not occur, it is possible to consider algebraic geometry as a special case of differential algebra.1
It is noteworthy that a subject so substantial as differential algebra owes its existence to one person. J. F. Ritt (1893-1951) was not only its founding father, but also its principal prophet and practitioner. Today, 22 years after his death, the majority of the main results, and the deepest ones, are due to him, and despite a new look, the mainlines of the subject today are the same as in 1951. It had already become clear then that differential algebra is pure algebra, and although Ritt's life blood was classical analysis, in his second book2 on the subject [95]! he made a great effort to meet the algebraist half way.
