We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur-Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data NevanlinnaPick interpolation problem in the Schur class on the unit polydisk.

For every bivariate polynomial p(z 1 , z 2

Abstract. We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from other equilibrium states is established. It enables us to gain insight into the structure of positive trigonometric polynomials and Dirichlet spaces associated with finitely atomic measures. We also derive a related family of linear second order differential equations with polynomial solutions.

For every multivariable polynomial p, with p(0) = 1, we construct a determinantal representationwhere Z is a diagonal matrix with coordinate variables on the diagonal and K is a complex square matrix. Such a representation is equivalent to the existence of K whose principal minors satisfy certain linear relations. When norm constraints on K are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial q, q(0) = 0, satisfies the von Neumann inequality, then 1 − q admits a determinantal representation with K a contraction. On the other hand, every determinantal representation with a contractive K gives rise to a rational inner function in the Schur-Agler class.for a d-variable polynomial p(z), z = (z 1 , . . . , z d ), with p(0) = 1. Here n = (n 1 , . . . , n d ) is in the set N d 0 of d-tuples of nonnegative integers, |n| = n 1 + · · · + n d , Z n = d i=1 z i I ni , and K is a complex square matrix. It is of interest of how and to what extent, the algebraic and operator-theoretic properties of the polynomial correspond to the size and norm of the matrix K of its representation.

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