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We consider a square random matrix of size $N$ of the form $P(Y,A)$ where $P$is a noncommutative polynomial, $A$ is a tuple of deterministic matricesconverging in $\ast$-distribution, when $N$ goes to infinity, towards a tuple$a$ in some $\mathcal{C}^*$-probability space and $Y$ is a tuple of independentmatrices with i.i.d. centered entries with variance $1/N$. We investigate theeigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where $c$ is acircular system which is free from $a$. We provide a sufficient condition toguarantee that these eigenvalues coincide asymptotically with those of$P(0,A)$.

Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices