Positive definite collections of disks Vladimir Tkachev 31A1530C1033C45potential theoryorthogonal polynomialspositivity Let $Q(z,w) = -\prod_{k=1}^{n}[(z - a_k)(\bar{w} - \bar{a}_{k}) - R^{2}]$. The main result of the paper states that in the case when the nodes $a_{j}$ are situated at the vertices of a regular $n$-gon inscribed in the unit circle, the matrix $Q(a_{i}, a_{j})$ is positive definite if and only if $R < \rho_{n}$, where $z = 2\rho_{n}^{2} - 1$ is the smallest $\ne -1$ zero of the Jacobi polynomial $\mathcal{P}^{n-2\nu, {-}1}_{\nu}(z)$, $\nu=[n/2]$. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.3004 10.1512/iumj.2006.55.3004 en Indiana Univ. Math. J. 55 (2006) 1907 - 1934 state-of-the-art mathematics http://iumj.org/access/