Cylindrical decomposition for systems transcendental in the first variable

Abstract

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p ( x , f 1 ( x ) , … , f m ( x ) , y 1 , … , y n ) , where p ∈ Q [ x , t 1 , … , t m , y 1 , … , y n ] and f 1 ( x ) , … , f m ( x ) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q [ x , f 1 ( x ) , … , f m ( x ) ] . In particular, the algorithm applies when f 1 ( x ) , … , f m ( x ) are real exp–log functions or tame elementary functions.