Necklaces count polynomial parametric osculants

TL;DR

This paper establishes a combinatorial count for polynomial parametric curves approximating complex analytic curves, linking the number of such curves to primitive necklaces, and extends the results to higher-dimensional hypersurfaces.

Contribution

It introduces a novel connection between polynomial osculants and primitive necklaces, providing a combinatorial enumeration method and partial solutions to existing conjectures.

Findings

Number of polynomial osculants equals the count of primitive necklaces.

The count is odd when degrees are equal and squarefree.

Results extend to hypersurfaces in higher dimensions.

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 $i$.