Let $f = f(z_1, \dots, z_n)$ be a power series in $n$ variables with coefficients in the $p$-adic integers $\mathbb{Z}_p$. Let $g(z_1, \dots, z_n) = f(pz_1, \dots, pz_n)$, so that $g$ converges on all of $\mathbb{Z}_p^n$. Under what conditions is the zero set of $g$ algebraic, meaning that it is cut out by polynomial equations?
So far, I know that if $n=1$, then the answer is always by the Weierstrass preparation theorem. This result allows one to factor $g(z_1)$ as $g(z_1) = h(z_1)u(z_1)$, where $h$ is a polynomial and $u$ is a nowhere vanishing power series with coefficients in $\mathbb{Z}_p$.
But when $n > 1$, the Weierstrass preparation theorem in multiple variables gives only a factorization $g(z_1,\dots, z_n) = h(z_1,\dots,z_n)u(z_1,\dots,z_n)$, where $h$ is a polynomial in $z_1$ with coefficients in the power series ring $\mathbb{Z}_p[[z_2,\dots,z_n]]$. This is not immediately enough to deduce that the zero locus of $g$ is cut out by polynomials. Can anything further be said when $n>1$? What about when $n=2$?
