  We consider the problem of geometrically approximating a complex analytic
curve in the plane by the image of a polynomial parametrization $t \mapsto
(x_1(t),x_2(t))$ of bidegree $(d_1,d_2)$. We show the number of such curves is
the number of primitive necklaces on $d_1$ white beads and $d_2$ black beads.
We show that this number is odd when $d_1=d_2$ is squarefree and use this to
give a partial solution to a conjecture by Rababah. Our results naturally
extend to a generalization regarding hypersurfaces in higher dimensions. There,
the number of parametrized curves of multidegree $(d_1,\ldots,d_n)$ which
optimally osculate a given hypersurface are counted by the number of primitive
necklaces with $d_i$ beads of color $i$.
