# Power series ring and monomials

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 $$\begin{equation*} P_{\alpha,\beta}(x,y) \colon= (\alpha + x)(\beta + y). \end{equation*}$$

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

No. For instance, take $$n=\epsilon=1$$, $$p=2$$. Then the only $$\epsilon$$-monomials are 1 and $$X_1$$ and the $$2\epsilon$$-monomials are 1, $$X_1$$, $$X_1^2$$ and $$X_1^3$$. Take $$\alpha=\beta=X_1$$, $$x=y=X_1^2-X_1$$. We have $$(\alpha+x)(\beta+y)=X_1^2X_1^2=X_1^4$$, which involves no $$2\epsilon$$-monomials.
• $x$ should not contain $\alpha$, as I understood the question May 8 '19 at 12:57
• Yes, $x$ should not contain $\alpha$, right. May 8 '19 at 15:36
This answer replacees my earlier, incorrect one. The answer is still "no", even if $$x$$ does not involve $$\alpha$$ and $$y$$ does not involve $$\beta$$. Example: take $$n=2$$, $$p=3$$, $$\alpha=\beta=X_1^2X_2^8$$, $$\epsilon=1$$, $$x=X_1^4$$, $$y=-X_2^{16}$$. We have $$(\alpha+x)(\beta+y)=X_1^6X_2^8-X_1^2X_2^{24}$$ which contains no $$2\epsilon$$-monomials.