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We prove that every matrix-valued rational function $F$, which is regular onthe closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ andwhich has the associated Agler norm strictly less than 1, admits afinite-dimensional contractive realization $$F(z)= D +C\mathbf{P}(z)_n(I-A\mathbf{P}(z)_n)^{-1} B. $$ Here $\mathcal{D}_\mathbf{P}$is defined by the inequality $\|\mathbf{P}(z)\|<1$, where $\mathbf{P}(z)$ is adirect sum of matrix polynomials $\mathbf{P}_i(z)$ (so that appropriateArchimedean and approximation conditions are satisfied), and$\mathbf{P}(z)_n=\bigoplus_{i=1}^k\mathbf{P}_i(z)\otimes I_{n_i}$, with some$k$-tuple $n$ of multiplicities $n_i$; special cases include the open unitpolydisk and the classical Cartan domains. The proof uses a matrix-valuedversion of a Hermitian Positivstellensatz by Putinar, and a lurking contractionargument. As a consequence, we show that every polynomial with no zeros on theclosure of $\mathcal{D}_\mathbf{P}$ is a factor of $\det (I -K\mathbf{P}(z)_n)$, with a contractive matrix $K$.

Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials