On the Exponents of Differential Ideals

Abstract

In the theory of polynomial ideals, in algebra, there are methods, stemming from the theorem of M. Noether, and associated with the names of E. Bertini, E. Lasker, F. S. Macauley, K. Hentzelt, H. Kapferer and P. Dubreil, for finding the exponent of an ideal, or at least a bound for the exponent. When one seeks to create a notion of exponent for of differential polynomials, one is forced, because of a situation revealed by H. W. Raudenbush,' to admit infinite exponents as well as finite ones. The investigation of such exponents, finite or infinite, to some extent for differential of a general character, and to a deeper extent for differential generated by a form in one unknown of the first order, is the object of the present paper. If we may refer to ?2 below for a definition of the exponent of a differential ideal, we shall proceed to enumerate our results. Part II, which presents what is possibly the most interesting portion of our work, deals with differential generated by a single form A, in one unknown, of the first order. The concept of multiplicity of a singular solution of A is introduced (?6) and it is shown (?7) that if A has a singular solution of multiplicity exceeding unity, then the differential ideal generated by A has exponent infinity. Forms A which have singular solutions, all of multiplicity unity, are discussed in ??8-10. Such singular solutions are divided into two classes, and, guided by a general theorem due to J. F. Ritt, we secure a decomposition of the differential ideal generated by A which puts these two classes into evidence (?8). In ?10 it is proved, under an additional assumption (regular type) that the exponent of the differential ideal generated by A is unity or two according as all the singular solutions are in the first class or at -least one singular solution is in the second class. The differential ideal generated by a form A which has no singular solutions, is shown, under a certain additional assumption, to have exponent unity (?11). Part II concludes with a discussion of a type of form which we call hyperelliptic. The intermediate ideals are found, and they are shown to fall into chains of a fixed length. Part III contains a brief discussion of chains of differential ideals. A theorem is proved which gives a bound for the exponent of a so-called principal chain in terms of the length of the chain (?14). Part I begins with a statement, in abstract form, of a decomposition theorem due to Ritt. After the definitions of relative exponent and other terms, there