  Methods from algebra and algebraic geometry have been used in various ways to
study linkages in kinematics. These methods have failed so far for the study of
linkages with helical joints (joints with screw motion), because of the
presence of some non-algebraic relations. In this article, we explore a
delicate reduction of some analytic equations in kinematics to algebraic
questions via a theorem of Ax. As an application, we give a classification of
mobile closed 5-linkages with revolute, prismatic, and helical joints.
