Let $f \in \mathbb{Z}[x_1, \ldots, x_n]$ and $p$ be a prime. Let $\nu_t(p)$ denote the number of solutions $\mathbf{x} \in ((\mathbb{Z}/p^t \mathbb{Z}))^*)^n$ to the congruence $$ f( \mathbf{x} ) \equiv 0 \ (\text{mod }{p^t}). $$ We define (something similar to the $p$-adic density) $$ \mu(p) = \lim_{t \rightarrow \infty} \frac{ p^t \nu_t(p) }{ \phi(p^t)^n }. $$ Could someone please explain how to show $$ \mu(p) > 0 $$ provided the equation $f( \mathbf{x} ) = 0$ has a non-singular solution in $\mathbb{Z}_p^{\times}$, the units of $p$-adic integers? Thank you very much!

PS I asked this question in mathstack exchange https://math.stackexchange.com/questions/1579876/showing-the-positivity-of-p-adic-density-of-zeroes-of-a-polynomial. I wasn't able to get an answer there so I thought I would try it on overflow.

PPS This came up when I was reading an article "Diophantine equations in the primes" http://link.springer.com/article/10.1007%2Fs00222-014-0508-1 I was interested in seeing the details but I couldn't quite figure it out.

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