Root separation on generalized lemniscates (Q5957964)

The problem of separating the roots of a complex polynomial with respect to a given planar curve is classical. Its applications to the stability of mechanical systems are well known and widely spread. The best separation algorithms, due to Routh, Hurwitz, Schur, Cohn, Lienart and Chipard refer to the upper half-plane or to the disk. A good reference for the history of this subject is \textit{M. G. Krein} and \textit{M. A. Naimark}, [Linear Multilinear Algebra 10, 265-308 (1981; Zbl 0584.12018)]. The class of quadrature domains was introduced in \textit{D. Aharonov}, and \textit{H. S. Shapiro} [J. Anal. Math. 30, 39-73 (1976; Zbl 0337.30029)], in connection with some extremal problems in conformal mapping theory. They are algebraic domains with a special, irreducible form, of their defining polynomial. They are dense among all planar domains, in the Hausdorff metric. The note under review exploits some recent progress in understanding the nature of the defining polynomial of a quadrature domain. As a consequence, a numerical test for a polynomial to have all roots outside a quadrature domain is obtained.