A characterization of complex plane Poncelet curves

We consider algebraic curves in the complex affine plane. A natural extension of the existing definition of Poncelet curves in the real plane to the complex plane is presented. Three equivalent polynomial equations in tangent coordinates are given for complex plane Poncelet curves: (a) the polynomial which generates the Bezoutian form with parameters-the foci of the curve; (b) the Darboux equation with parameters-the vertices of a Poncelet polygon; (c) the determinant equation involving matrices having certain specific properties. We use these polynomials in order to solve Poncelet-type problems. Namely, criteria are proved for real Poncelet curves to be generated by matrices that admit unitary bordering. These criteria answer the question when a convex Poncelet curve which is inscribed in a convex polygon is the boundary of a numerical range of a matrix.We also demonstrate that the general theorems of the first three sections may shorten the proofs of some known results.