## Geometry Seminar

**Date:** November 17, 2017

**Time:** 4:00PM - 5:00PM

**Location:** BLOC 628

**Speaker:** Taylor Brysiewicz, TAMU

**Title:** *Counting polynomially parametrized interpolants via Necklaces*

**Abstract:** We consider the problem of locally approximating an analytic curve in the complex plane plane by a polynomial parametrization t -> (x_1(t),x_2(t)) of bidegree (d_1,d_2). Contrary to Taylor approximations, these parametrizations can achieve a higher order of contact at the cost of losing uniqueness and possibly the reality of the solution. We study the extent to which uniqueness fails by counting the number of such curves as the number of aperiodic combinatorial necklaces on d_1 white beads and d_2 black beads. We analyze when this count is odd as an initial step in studying when real solutions exist.