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Decision problems for differential equations

Published online by Cambridge University Press:  12 March 2014

J. Denef  and
L. Lipshitz
J. Denef
Affiliation:
Department of Mathematics, Catholic University of Leuven, B-3030 Leuven, Belgium
L. Lipshitz
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
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Extract

In this paper we shall consider some decision problems for ordinary differential equations. All differential equations will be algebraic differential equations, i.e. equations of the form P(x, y, y′, …, y(n)) = 0 (or P(x, y1, …, ym, y′1 …, y′m, …) = 0 in the case of several dependent variables), where P is a polynomial in all its variables with rational coefficients. (We call P a differential polynomial.)

Jáskowski [6] showed that there is no algorithm to determine if a system of algebraic differential equations (in several dependent variables) has a solution on [0, 1]. An easier proof of this using results on Hilbert's tenth problem is given in [2]. It is natural to restrict the problem in the hope of finding something which is decidable. Two ways to do this are the following: (a) One can ask only for the existence of solutions locally, say around x = 0. Here solution may mean (i) analytic functions, (ii) germs of C∞ functions, or (iii) formal power series, (b) One can consider only one equation in one dependent variable, but ask for the existence of a solution on an interval. In [4] we gave an algorithm for deciding (a)(ii) and (a)(iii) and showed that (a)(i) is undecidable. In §4 (Theorem 4.1) we shall show that (b) (for real analytic solutions) is undecidable. This partially answers Problem 9 of Rubel [13], which stimulated this investigation. We shall also prove (Theorem 4.2) that, given an analytic function f(x) which is the solution of an algebraic differential equation, determined by some initial conditions at x = 0, one cannot in general determine if the radius of convergence of f is < 1 or ≥ 1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 941 - 950
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Aberth, O., The failure in computable analysis of a classical existence theorem for differential equations, Proceedings of the American Mathematical Society, vol. 30 (1971), pp. 151–156.CrossRefGoogle Scholar
[2]Adler, A., Some recursively unsolvable problems in analysis, Proceedings of the American Mathematical Society, vol. 22 (1969), pp. 523–526.CrossRefGoogle Scholar
[3]Davis, M., Matijasevič, Y. and Robinson, J., Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution, Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, vol. 28, American Mathematical Society, Providence, Rhode Island, 1976, pp. 323–378.CrossRefGoogle Scholar
[4]Denef, J. and Lipshitz, L., Power series solutions of algebraic differential equations, Mathematisehe Annalen, vol. 267 (1984), pp. 213–238.CrossRefGoogle Scholar
[5]Henson, C. W. and Rubel, L. A., Some applications of Nevanlinna theory to mathematical logic; identities of exponential functions, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 1–31.Google Scholar
[6]Jaśkowski, S., Example of a class of systems of ordinary differential equations having no decision method for existence problems, Bulletin de l'Académie Polonaise des Sciences, Classes III, vol. 2 (1954), pp. 155–157.Google Scholar
[7]Macintyre, A., The laws of exponentiation, Model theory and arithmetic (Paris, 1979), Lecture Notes in Mathematics, vol. 890, Springer-Verlag, Berlin, 1981, pp. 185–197.CrossRefGoogle Scholar
[8]Mahler, K., Lectures on transcendental numbers, Lecture Notes in Mathematics, vol. 546, Springer-Verlag, Berlin, 1976.CrossRefGoogle Scholar
[9]Pour-El, M. and Richards, I., A computable ordinary differential equation which possesses no computable solution, Annals of Mathematical Logic, vol. 17 (1979), pp. 61–90.CrossRefGoogle Scholar
[10]Pour-El, M. and Richards, I., Noncomputability in mathematics and physics: a complete determination of the class of noncomputable linear operators, Advances in Mathematics, vol. 48 (1983), pp. 44–74.CrossRefGoogle Scholar
[11]Richardson, D., Some undecidable problems involving elementary functions of a real variable, this Journal, vol. 33 (1968), pp. 514–520.Google Scholar
[12]Richardson, D., Solution of the identity problem for integral exponential functions, Zeitschrift für Mathematisehe Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 333–340.CrossRefGoogle Scholar
[13]Rubel, L., Some research problems about algebraic differential equations, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 43–52.CrossRefGoogle Scholar
[14]Scarpellini, B., Zwei unentscheidbare Probleme der Analysis, Zeitschrift für Mathematisehe Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 265–289.CrossRefGoogle Scholar
[15]van den Dries, L., Exponential rings, exponential polynomials and exponential functions, Pacific Journal of Mathematics, vol. 113 (1984), pp. 51–66.CrossRefGoogle Scholar