Effective Methods in Algebraic Geometry pp 427-440 | Cite as

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

## Abstract

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.

### Keywords

Manifold Mora Vica## Preview

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