Necklaces count polynomial parametric osculants

Abstract

We consider the problem of geometrically locally approximating a general complex analytic curve in the plane at a point by the image of a polynomial parametrization t↦(x1(t),x2(t)) of bidegree (d1,d2). We show the number of such approximating curves is the number of primitive necklaces on d1 white beads and d2 black beads. We show that this number is odd when d1=d2 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 (d1,…,dn) which optimally osculate a given hypersurface are counted by the number of primitive necklaces with di beads of color i. The proofs of these results give rise to a numerical homotopy algorithm for computing all multidegree (d1,…,dn) osculants to a general hypersurface at a point.