Some research problems about algebraic differential equations II

Abstract

Twenty-four new research problems are posed, and their background and partial solutions are sketched. Many of these problems are in the (somewhat unexpected) area of interaction between algebraic differential equations, topology, and mathematical logic. Introduction. What has struck me most about the subject of algebraic differential equations (ADEs) is how easy it is to pose hard problems. Another way of saying the same thing is that the subject is still in a formative state, with many fundamental problems not even posed, much less solved. I am going to present the first part (Problems 1-3) of this paper as I first wrote it. Thereis a rather dramatic twist at its end, as you will see. Explanatory section. An algebraic differential equation (ADE) is just one where the left-hand side is a polynomial in everything. More specifically, let P(x0, xl,... ,xn+,) be a polynomial in n + 2 variables, not the zero polynomial, with complex coefficients, or with real coefficients according to the context. Then the equation (1) P(z, w(z), W'(Z),. . . ,W(n)(Z)) = 0 is called an ADE. We may write this as (1') P (Z, W)=O. In the real case, we prefer to write (2) P(x, y(x), y'(x),. . . ,y(n)(x)) = 0 or just (2') P(x, y) =0. In both of these contexts, we call P a differentialpolynomial. What we mean by a solution of an ADE depends somewhat on the context. In the complex case it will mean an analytic (or sometimes a meromorphic) function on a region S2 that satisfies (1). When this happens, we call w(z) a hypotranscendental function. If w(z) satisfies no ADE, then it is assigned the beautiful adjective transcendentally transcendental. (See [OST] for a good introduction to this subject. In Received by the editors January 14, 1982. 1980 Mathematics Subject Classification. Primary 34-XX, 34A34. The research of the author was partially supported by a grant from the National Science Foundation. ?1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page