Necklaces count polynomial parametric osculants. (March 2021) - Brysiewicz, Taylor
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 ↦ ( 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.