Computing the Topology of a Bounded Non-Algebraic Curve in the Plane

Abstract It is shown that within a coordinate aligned rectangle in the real plane there exists a cylindrical decomposition of the zero set of any function of two variables which can be written as a polynomial in x, ex, y, ey. Also the construction of such a cylindrical decomposition is Turing reduced to problems involving definable constants. In this way the topology of the zero set of such a function is computable in a bounded coordinate aligned rectangle, depending on an oracle to solve the constant problem. This result is then generalized to functions which can be written as polynomials in x, y and any number of basic functions in x or in y which belong to a large class of analytic functions, including all those which satisfy first order algebraic differential equations. The techniques used are false derivatives and local Sturm sequences.