Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a formal power series ring over a field $K$ of characterisc $p > 0$ in $n$ variables.

For a given positive number $\epsilon > 0$ we call a monomial $X_{i_1}^{e_{i_1}} \cdots X_{i_n}^{e_{i_n}}$ an $\epsilon$-monomial if it satisfies the following conditions simultaneously$\colon$
\begin{align*}
&
e_{i_1}/p^{i_1} < \epsilon \\
&
e_{i_2}/p^{i_2} < \epsilon \\
&
\cdots \\
&
e_{i_n}/p^{i_n} < \epsilon.
\end{align*}
Let $\alpha$, $\beta$ be two $\epsilon$-monomials. Then for two elements $x, y \in (X_1,\ldots,X_n)$, i.e., the unique maximal ideal of $A_n$, we consider the product defined by
\begin{equation*}
P_{\alpha,\beta}(x,y) \colon= (\alpha + x)(\beta + y).
\end{equation*}

## Q. Suppose that $x$ (resp. $y$) comprises monomials different from $\alpha$ (resp. $\beta$). That is, $x$ (resp. $y$) is composed of monomials neither of which is equal to $\alpha$ (resp. $\beta$).

## Then does the product $P_{\alpha, \beta}(x, y)$ always contain a non-zero $2\epsilon$-monomial?