Poncelet-Darboux Curves, their Complete Decomposition and Marden Theorem

Abstract

Poncelet–Darboux curves are studied. A Poncelet–Darboux curve of degree n−1 is completely decomposable if it consists of (n−1)/2 conics if n is odd; if n is even, completely decomposable curve consists of (n−2)/2 conics and a line. An effective criterion for complete decomposability in terms of pair of polynomials which defines a Poncelet–Darboux curve is derived. A relationship between Poncelet–Darboux curves and the Marden theorem from geometric theory of zeros of polynomial and rational functions is established.