Necklaces count polynomial parametric osculants

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 bar right arrow (x(1)(t), x(2)(t)) of bidegree (d(1), d(2)). We show the number of such approximating 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), ..., d(n)) which optimally osculate a given hypersurface are counted by the number of primitive necklaces with d(i) beads of color i. The proofs of these results give rise to a numerical homotopy algorithm for computing all multidegree (d(1), ..., d(n)) osculants to a general hypersurface at a point. (C) 2019 Elsevier Ltd. All rights reserved.