Finding roots of equations involving functions defined by first order algebraic differential equations

  • Daniel Richardson
Part of the Progress in Mathematics book series (PM, volume 94)


A method is given for approximating solutions of f (x) = 0, for f in a certain class F of real valued analytic functions of one real variable. The method depends on being able to decide the sign of f (r) for given f in F and rational r. That is, approximation of roots is Turing reduced to the constant problem. The last root problem is evaded: essentially solutions are only found in given finite intervals.

The class of functions F contains x, exp(x), log(x) for x > 0, sin(x) for x in (—π/2, π/2), and is closed under field operations, differentiation and integration. F is built up by successively adding solutions g of first order differential equations q(x, g, g′) = 0, where q is a polynomial whose coefficients may involve previously defined functions.

The main technique used is construction of a false derivative. A false derivative of f (x) is a function f *(x) which is continuous and has the same sign as f′(x) whenever f(x) = 0.


Manifold Mora Vica 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Arnon, D. S., and B. Buchberger, (Editors), “Algorithms in real algebraic geometry,” Academic Press, 1988.Google Scholar
  2. [2]
    Davenport, J. H., Computer Algebra for Cylindrical Decomposition, University of Bath computer science technical report 88-110.Google Scholar
  3. [3]
    van den Dries, L., On the elementary theory of restricted elementary functions, The Journal of Symbolic Logic 53 (1988), 769–808.Google Scholar
  4. [4]
    van den Dries, L. and J. Denef, P-adic and real subanalytic sets, Annals of Mathematics 128 (1988), 79–138.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Khovanskii, A. G., Fewnomials and Pfaff manifolds, in “Proc. Int. Congress of Mathematicians,” Warsaw 1983, pp. 549–564.Google Scholar
  6. [6]
    Khovanskii, A.G., On a class of systems of transcendental equations, Soviet Math. Dokl. 22 (1980), 762–765.Google Scholar
  7. [7]
    Lombardi, H., Algebre Elementaire en Temps Polynomial, Thesis, University of Nice, 1989.Google Scholar
  8. [8]
    Richardson, D., Wu’s method and the Khovanskii finiteness theorem, Bath Computer Science, technical report 89-29.Google Scholar
  9. [9]
    Shackell, J., Zero-Equivalence in function fields defined by algebraic differential equations.Google Scholar
  10. [10]
    Shackell, J., A Differential-equations approach to functional equivalence.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Daniel Richardson
    • 1
  1. 1.Department of MathematicsUniversity of BathUK

Personalised recommendations