  In this article we give an explicit algorithm which will determine, in a
discrete and computable way, whether a finite piecewise Euclidean complex is
non-positively curved. In particular, given such a complex we show how to
define a boolean combination of polynomial equations and inequalities in real
variables, i.e. a real semi-algebraic set, which is empty if and only if the
complex is non-positively curved. Once this equivalence has been shown, the
main result follows from a standard theorem in real algebraic geometry.
