Necklaces count polynomial parametric osculants

Abstract

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 $i$.