Ritt, Joseph Fels
RITT, JOSEPH FELS
After two years of study at the College of the City of New York, Ritt obtained the B.A. from George Washington University in 1913. He received the Ph.D. from Columbia University in 1917 for a work on linear homogeneous differential operators with constant coefficients. He was colloquium lecturer of the American Mathematical Society (1932), a member of the National Academy of Sciences, and vice-president of the American Mathematical Society from 1938 to 1940.
Ritt’s early work was highly classical. Papers entitled “On Algebraic Functions Which Can Be Expressed in Terms of Radicals,” “Permutable Rational Functions,” and “Periodic Functions With a Multiplication Theorem” were the. result of a thorough study of classic masters. His work on elementary functions took its inspiration directly from Liouville.
Ritt also investigated the algebraic aspects of the theory of differential equations, considering differential polynomials or forms in the unknown functions y1 ,…, yn. and their derivatives with coefficients that are functions meromorphic in some domain. Given a system σ of such forms, he shows that there exists a finite subsystem of σ having the same set of solutions as σ. Furthermore, if a form G vanishes for every solution of the system of forms H1 ,…, Hr , then some power of G is a linear combination of the Hi and their derivatives. These arguments lead to the statement that every infinite system of forms has a finite basis.
In considering reducibility Ritt concluded that the perfect differential ideal generated by a system of forms equals the intersection of the prime ideals associated with its irreducible components. The purpose of this work was to advance knowledge of “general” and “singular” solutions, which in the preceding literature (Laplace, Lagrange, and Poisson, for example) had been very unsatisfactory.
Contributions to algebraic differential equations and algebraic difference equations were made by many of Ritt’s students after 1932, in particular, by E. R. Kolchin, W. Strodt, H. W. Raudenbush, and H. Levi. Differential algebra, a new branch of modern algebra, also owes much to these early researches.
The most important aspects of Ritt’s work are summed up in two books published in the Colloquium Publications series of the American Mathematical Society: Differential Equations From the Algebraic Standpoint (New York, 1932) and Differential Algebra (New York, 1950).
Complete bibliographies of Ritt’s writings are included in the notices by E. R. Lorch, in Bulletin of the American Mathematical Society, 57 (1951), 307–318; and by Paul A. Smith, in Biographical Memoirs. National Academy of Sciences, 29 (1956), 253–264.
Edgar R. Lorch