Let $f(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ and consider the equation $f(\mathbf{x}) = 0$. I am wondering what exactly does it mean by "the equation $f(\mathbf{x}) = 0$ has a non-singular solution in the units of $p$-adic integers $\mathbb{Z}_p^{\times}$"?
I encountered this in an article and I thought I knew what it meant but I had some second thoughts and I wanted to verify. This is what I thought: There exists $(z_1, ..., z_n) \in (\mathbb{Z}_p^{\times})^n$ such that it solves the equation and that $$ \frac{\partial}{\partial x_i} f(\mathbf{x}) |_{\mathbf{x} = (z_{1,m}, ..., z_{n,m})} \not \equiv 0 \pmod{ p^m } $$ for some $1 \leq i \leq n$ and some $m \geq 0$. Here I am using the inverse limit representation of $z_i = (z_{i,0}, z_{i,1}, ...)$.
I would greatly appreciate if someone could possibly tell me the correct definition in case what I thought was not true. Thank you very much!
Modulo my inability to read your subscripts, it looks right to me, but I’d say (equivalently, I think) that there’s an $n$-tuple $\mathbf x$ of elements of $\Bbb Z_p^*$ such that $f(\mathbf x)=0$ and an $i$ such that $f_i(\mathbf x)\ne0$, where $f_i$ is the derivative with respect to the $i$-th named variable. (It’s always a mess to notate the evaluation of a partial derivative at a point with the $\partial$ notation, and occasionally ambiguous.)