Finite-dimensional contractive realizations on polynomially defined domains and determinantal representations of multivariable polynomials

We prove that every matrix-valued rational function F, which is regular on the
closure of a bounded domain DP in Cd and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization
F(z) = D + CP(z)n(I − AP(z)n) −1B.
Here DP is defined by the inequality kP(z)k < 1, where P(z) is a direct sum of matrix polynomials Pi(z) (so that an appropriate Archimedean condition is satisfied), and P(z)n = Lk i=1 Pi(z)⊗Ini , with some k-tuple n of multiplicities ni; special cases include the open unit polydisk and the classical Cartan domains of types I–III. The proof uses a matrix-valued version of a Hermitian Positivstellensatz by Putinar, and a lurking contraction argument. As a consequence, we show that every polynomial with no zeros on the closure of DP is a factor of det(I − KP(z)n), with a contractive matrix K.
When DP is the open unit polydisk, we show that a polynomial with no zeros in the domain is the denominator of a rational inner function of the Schur–Agler class if and only if it admits a contractive determinantal representation up to an almost self-reversive factor.
We also show that every rational inner function which is regular on the closed unit polydisk can be multiplied with another such function so that the product is in the Schur–Agler class.