Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.

Suppose we have two elements $\alpha = X_1^{e_1} \cdots X_n^{e_n}$, $\beta = Y_1^{f_1} \cdots Y_n^{f_n}$ such that some of $e_i$ and some of $f_i$ are bigger than $0$.

We add $A \in {\frak m}_{A_n}$ to $\alpha$ and $B \in {\frak m}_{A_n}$ to $\beta$ such that $A$ (resp. $B$) comprises scalar times monomials different from $\alpha$ (resp. $\beta$).

Consequently, $A + \alpha \in {\frak m}_{A_n}$ (resp. $B + \beta \in {\frak m}_{A_n} $) contains $\alpha$ (resp. $\beta$) as its constituent.

Q. Does the multiplication $(A + \alpha)(B + \beta)$ always contain nonzero monomial $X_1^{g_1} \cdots X_n^{g_n}Y_1^{g_{n+1}} \cdots Y_n^{g_{2n}}$ up to scalar coefficient such that inequalities $g_1 \leq 2e_1$,...,$g_n \leq 2e_n$, $g_{n+1} \leq 2f_1$, ..., $g_{2n} \leq 2f_n$ hold simultaneously?

  • $\begingroup$ What happens if you take the initial terms in degree lexicographic order? $\endgroup$ – Zach Teitler Jun 3 at 4:23
  • $\begingroup$ What do you mean by taking the initial term? For example, you mean $\alpha = X_1^e, \beta = Y_1^f$ ? $\endgroup$ – Rinmyaku Jun 3 at 8:34

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