Functions satisfying elementary relations

In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form R = 0 where R is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations. Introduction. The simplest case of determining when a solution of a differential equation satisfies an elementary relation is that of the single equation y' = o where a is an elementary function, that is, a function expressible in terms of a finite number of algebraic operations, logarithms, and exponentials. We are then asking, when does the integral y of a satisfy some elementary relation (for example, exp(.y1/2) + z2 + (log(log y)) • (exp zy) = 0)? In [4], [5], Ritt answered this question by showing that if y satisfies an elementary relation, it must then actually be elementary. This implies, for example, that no elliptic integral satisfies an elementary relation. In Theorem 1, we generalize this result to consider the collection of equations y\ = a,,... ,y'„ — an where the a, are in some differential field K, and show that if the solutionsyx,... ,yn satisfy an elementary relation (such as^exptTT log.y2)) + z\y\ 1)I/2 = 0), then there is some nontrivial linear combination c,yx + • • • + c„yn, with the c,'s constant, which is elementary with respect to K. With this result, we are able to show that certain classes of integrals satisfy no elementary relations. The next case to consider is that of a first order differential equation F(z,y,y') = 0, where F is a polynomial in y and y' with coefficients in C(z). This case was originally treated by Mordukhai-Boltovski [4]. He showed that if a solution y of such an equation satisfies an elementary relation, then it satisfies a relation of the form c, logcpx(z,y) + • • • + cn \ogyn(z,y) + <p0(z,y) = 0, Received by the editors November 27, 1974. AMS (MOS) subject classifications (1970). Primary 12H05. O American Mathematical Society 1977