UB-matrices and conditions for Poncelet polygon to be closed Original Research Article

A sufficient condition for the generation of a package of algebraic Poncelet curves is presented. The condition is obtained via the link between the parameters of the nested Poncelet curves and the eigenvalues of matrices which admit unitary bordering. This condition results in an example refuting the conjecture raised in [Linear and Multilinear Algebra 45 (1998) 49; Linear Algebra Appl. 329 (2001) 61] that algebraic Poncelet curves are always generated by such matrices. Then the condition is applied to the case of nested quadrics in order to determine whether they form the Poncelet configuration, i.e., whether there exists a closed polygon inscribed in the outer quadric and circumscribed about the inner one. The classical Cayley criterion answers this question for a given number N of sides of a polygon. In this paper, a different kind of criterion in the form of rational recurrent equations is developed. Moreover, sufficient conditions are derived for the case when a closed Poncelet polygon with any number of sides does not exist.