Foreword from the guest editors

This special issue on Differential Algebra and Differential Equations was the idea of Bob Caviness and Bruno Buchberger and originated from a special year in differential algebra that was organized by Raymond Hoobler and William Sit at The City College of New York during 1995–1996. It may interest the reader to recall the research environment under which the special year was started. As is well known, the theory of Gröbner basis, which deals with systems of algebraic equations and originally developed by Buchberger, was already widely applied in symbolic computations by 1994. This period of research, spanning more than two decades since the 1970s, coincided with a new revolution in computer technology, and rapid advances in computers made this famous algorithm practical for many problems. In contrast, the theory of differential algebra, which deals with systems of algebraic differential equations and originally developed by Joseph F. Ritt and Ellis R. Kolchin, was known only to very few researchers in symbolic computation. Later, the methods and theory in differential algebra, refined and generalized in the 1950s by Abraham Seidenberg, Azriel Rosenfeld, and Howard Levi (to name just a few others besides the two founders), were described in Kolchin (1973). However, the algorithms are not practical for many general classes of differential algebraic systems because of their complexity. In addition, they call for analogous algorithms in algebraic geometry, and several important open problems remain. Differential algebra has always been an obscure subject. The technique of characteristic sets introduced by Ritt for differential algebraic systems was brought to the attention of the symbolic computation community in the late 1970s and early 1980s by Wen-Tsün Wu, who pioneered the method to solve problems in mechanical theorem proving for Euclidean geometry (see Chou, 1988). Before this, the only part of differential algebra that was known outside Kolchin’s sphere (and a few model theorists perhaps) was differential Galois theory, which originated from and extended the works of Charles Picard and Ernest Vessiot in the nineteenth century. This theory parallels the Galois theory for an algebraic equation and studies a differential equation and the properties of its solutions, in particular, their solvability in quadratures, through its differential Galois group. The popularity of the Picard–Vessiot theory was largely due to an easy to read little monograph by Irving Kaplansky (1957). Well before the present advance of computers, differential algebraists have been creating newer theories and developing algorithms (in principle). Established in the 1930s, the Ritt–Raudenbush Basis Theorem (Ritt, 1932; Raudenbush, 1934) for radical differential ideals is a close analog of the Hilbert Basis Theorem (the exact analog is not true). Every radical differential ideal is also a finite, irredundant intersection of prime differential ideals. These two results, and many others as shown in Kolchin (1973), can be generalized to partial differential polynomial rings in several differential indeterminates over differential fields of arbitrary characteristics. During the next thirty or so years,