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Suppose $P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1})$ is a noncommutative polynomial with positive coefficients. We may then consider the map $g:U(n)^k\rightarrow\mathbb{C}$ from the unitary group $U(n)$, given by $(A_1,\dots,A_k)\mapsto \det(P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1}))$. How can I show that (for large $n$), with high probability (with respect to the Haar measure on $U(n)^k$), $|g|\geq\frac{1}{2^{n^2}}$? We may change $2^{n^2}$ if necessary. I want to show that the determinant is bounded away from $0$ as a function of $n$ for large $n$, asymptotically almost surely.

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  • $\begingroup$ What if the positive coefficients are themselves very small? You need some condition to rule this case out. $\endgroup$
    – Will Sawin
    Jan 29, 2022 at 13:49
  • $\begingroup$ Let's say the coefficients are all $1$. However, would small coefficients (the polynomial is fixed) cause a problem if the question is for large enough $n$? $\endgroup$
    – user447643
    Jan 29, 2022 at 14:00
  • $\begingroup$ Ah - surely not after possibly changing the expression $2^{n^2}$ - if it is true for coefficients $\geq 1$, it is true for constant coefficients with any faster-decreasing lower bound. $\endgroup$
    – Will Sawin
    Jan 29, 2022 at 14:04

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