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Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that $\xi$ is trivial on $p^v\mathbb{Z}_p^n$, but not on $p^{v-1}\mathbb{Z}_p^n$. The question is:

Find an upper bound of $\lvert\hat{g}(\xi)\rvert$ in terms of $v(\xi)$.

If $f$ is a polynomial in one variable, then I get

$\lvert\hat{g}(\xi)\rvert \ll p^{-v(\xi)}$

The solution is quite simple: One solves the problem for f(z)=z, in which case $\hat g(\xi)$ can be explicitly computed. The general case then follows easily, because we can factor f into linear factors times a non-vanishing function. This solution, however, doesn't work in higher dimensions.

My guess is that the same bound holds for higher dimensions, but I still haven't managed to show that. Has this problem already been solved? Do you know of any books or papers that might be helpful? Any help would be appreciated. Thanks.

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