For the generic 3R robot manipulator (3 serial links connected by revolute joints), the equations describing the position of the end-effector (hand) in terms of the joint angles which are normally written as a system of 3 nonlinear equations (one for x, y, and z in the workspace) can be condensed into a single quartic in terms of one of the joint angles. For this case, one can for example create this polynomial in terms of $\tan(\theta_3/2)$. In Kohli & Spanos, Workspace Analysis of Mechanical Manipulators Using Polynomial Discriminants, they write:
The determination of the joint variables for a specified position of the hand (last link) is conducted by obtaining a set of equations relating the joint variables and constant kinematic parameters of the manipulator linkage to the hand position variables. In general, this set of equations is reduced to one equation of polynomial form in only one joint variable.
seemingly claiming that this can be done for higher degree of freedom (DOF) serial manipulators as well. Is this always the case, i.e. can you convert the 3 nonlinear equations describing forward kinematics into a single polynomial in one variable? And if so, is there a generic method for doing so? For a 6 DOF robot, this polynomial would be up to 16-th order.
The closest to a derivation for the 3 DOF case I found is in Using Computer Algebra Tools to Classify Serial Manipulators, but I can't figure out how they constructed the quartic.