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.