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The action of the rotation group $SO(3)$ on systems of $n$ points in the$3$-dimensional Euclidean space $\mathbf{R}^3$ induces naturally an action of$SO(3)$ on $\mathbf{R}^{3n}$. In the present paper we consider the followingquestion: do there exist $3$ polynomial functions $f_1$, $f_2$, $f_3$ on$\mathbf{R}^{3n}$ such that the intersection of the set of common zeros of$f_1$, $f_2$, and $f_3$ with each orbit of $SO(3)$ in $R^{3n}$ is nonempty andfinite? Questions of this kind arise when one is interested in relative motionsof a given set of $n$ points, i.e., when one wants to exclude the local motionsof the system of points as a rigid body. An example is the problem of decidingwhether a given polyhedron is non-trivially flexible. We prove that suchfunctions do exist. To get a necessary system of equations $f_1=0$, $f_2=0$,$f_3=0$, we show how starting by choice of a hypersurface in$\mathbf{CP}^{n-1}$ containing no conics, no lines, and no real points one canfind such a system.

Elimination of parasitic solutions in theory of flexible polyhedra