A differentially algebraic elimination theorem with application to analog computability in the calculus of variations

An elimination theorem is proved in differential algebra, from which it follows that an analytic solution of virtually any ordinary differential equation that you can "write down" must actually solve an algebraic differential equation. As a corollary, it follows that the solutions of a large class of variational problems can be produced by an analog computer. A hanging chain hangs in a catenary—essentially y = cosh x, which satisfies the algebraic differential equation y'2 y2 + 1 = 0. It is similar for other classical problems in the calculus of variations in one independent variable. One could ask: Is this an accident? Why doesn't the chain hang as y = T(x) or some other "bad" function, which by [HOL] satisfies no algebraic differential equation? In this paper we give the reason by proving a general theorem (see Theorem 2). This theorem is a direct consequence of an elimination theorem for differential equations (see Theorem 1) which implies, for example, that an analytic solution of y'y" +(x2 + l)sin ey + tan y' \ , „, -—H +J0(x+y") = -log* f" (where J0 is the Bessel function) must actually satisfy some algebraic differential equation (ADE), i.e. an equation of the form P(x, y, y ,. .,y{ ') = 0, where F is a polynomial in all its variables. This result answers, in much greater generality than stated, Problem 12 of [RUB]. In a certain sense then, for virtually every ordinary differential equation that you can "write down," its solutions satisfy an algebraic differential equation. Now the Shannon-Pour-El Thesis (see [SHA and POE], but see also [LIR] for further remarks) says essentially that a function y(x) is the output of a generalpurpose analog computer if and only if it is differentially algebraic (DA), which means that it satisfies some ADE. We shall call such a function "analog-computable. " (We need no new abbreviation—"DA" will suffice.) Received by the editors September 26, 1984 1980 Mathematics Subject Classification. Primary 34A34. 1 Supported by the National Science Foundation. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $. 25 per page 653 License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use 654 LA. RUBEL AND M. F. SINGER When it comes to (smooth) functions/of several variables xx,. ,xk, we say that / e DA if it is DA as a function of each variable when the others are held fixed. Ostrowski has shown [OST] that f(xx,. .,xn) is DA if and only if it satisfies an algebraic Mayer system of partial differential equations. This means that there are a certain number of the (mixed) partial derivatives of/, say px,. ,ph such that if p is any (mixed) partial derivative of/, then there is a rational function H such that P = Hp(xx,.,xn:px,.,Pl). For example, a + iï = 0 fL.0 dx2 dy2 dxdy is an algebraic Mayer system. (It is also easy to show (see [OST]) that f(xx,.,xn) is a DA function of its variables xx,. ,xn if and only if, for all choices xx(t),. ,xn(t) of DA functions of one variable, F(t) = f(xx(t),. ,xn(t)) is a DA function of one variable. Again, by [OST], it follows that sums, products, differences, quotients, compositions, inverses, derivatives, and integrals of DA functions are again DA. In other words, the usual processes of algebraic analysis keep one within the class of differentially algebraic functions. ) So another way to phrase our Theorem 2 is that if we have a problem in the calculus of variations in one dimension, and if the integrand of the integral to be minimized is analog-computable, then the solution is also analog-computable. Whether this carries over to higher dimensions (where the constraints are also supposed to be analog-computable) is an open problem that seems difficult. One ramification of this is the open problem, related to the Dirichlet problem for Laplace's equation in the disc. Problem. Suppose an are real (or complex) numbers with |a„| < 1/«! for n = 0,1,2, 3,. and suppose that £a„cos nx satisfies an algebraic differential equation. Must then Ea„ sin nx satisfy a (presumably different) algebraic differential equation? A second consequence of our elimination theorem is a differentially algebraic analog of the theorem of Weierstrass which considers meromorphic functions W that satisfy an addition formula of the form (*) Wiu + v) = AiWiu),Wiv)), where A is an algebraic function of two variables. He proved that W must be either (a) an algebraic function, (b) an algebraic function of some eXu, or (c) an algebraic function of some elliptic function. (See [ASK, Lecture 4] for this and related material.) We prove, in Theorem 3, that a meromorphic function W satisfies a formula (*) where A is differentially algebraic if and only if W is differentially algebraic. It is challenging to ask what happens when this restriction on A is strengthened to "linearly differentially algebraic," which means that, for every fixed t, A(s, t) satisfies, in s, a linear differential equation with polynomial coefficients, with a similar condition on A(s, t ) as a function of t for fixed s. With these preliminaries done, we may begin the body of the paper. License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use DIFFERENTIALLY ALGEBRAIC ELIMINATION THEOREM 655 Theorem 1. Let G(t_x, t0, tx,. .,tn) be a nontrivial analytic differentially algebraic function of its n + 2 variables, and let y(x) be an analytic solution, say, on an interval I, of the differential equation (1) Gix,yix),y'(x),. y)ix)) = 0. Then y(x) satisfies some algebraic differential equation on I. Proof. For simplicity, we will handle the first-order case where (1) reads (2) Gix,yix),y'ix)) = Q. The general case goes the same way, but with more complicated notation. We may suppose that there is a subinterval 7 ' of 7 throughout which the séparant, S = 9G/o\y' evaluated along our solutiony(x), does not vanish: dG/dy' =/= 0. (For if dG/dy' = 0 in an interval, then we could replace G by dG/dy'. We then have to worry about 32G/3(y')2 = 0 in a subinterval. Proceeding stepwise in this fashion, we arrive at an expression whose séparant does not vanish, unless there is a point where 8"c7/9(y')" = 0 for all n. But then, by analyticity, G is independent of y' (the highest-order derivative) and we are reduced to the case of a lower-order differential equation. So an argument by induction on the order completes the proof. ) Let us write (A0) Gix,yix),y'ix)) = 0, (A,) —Gix,yix),y'ix)) = 0. By the introductory remarks about algebraic Mayer systems, there are finitely many (mixed) partial derivatives <px,. ,<pk of G on which all the (mixed) partial derivatives of G rationally depend. Now in (A¡), perform the indicated differentiation by the chain rule, to get (a'i) £+. £*'+ *. "roÍ and solve for^" to get