About the exchange of $\sum$ and LM Given $f_i,g_i\in k[x_1,\cdots,x_n],1\leq i\leq s$, fix a monomial order on $k[x_1,\cdots,x_n]$, I was wondering whether there is an effective criterion to judge if this holds,$$\text{LM}(\sum_{i=1}^sf_ig_i)=\sum_{i=1}^s\text{LM}(f_ig_i),$$
where LM( ) is the leading monomial with respect to the fixed monomial order defined as follows,
$$\text{LM}(f)=x^{\text{multideg}(f)}.$$
And $\text{multideg}(f)=\text{max}(\alpha\in\mathbb Z_{\geq 0}^{n}:a_{\alpha}\neq0),$ where $f=\sum_{\alpha}a_{\alpha}x^{\alpha}.$

In characteristic 0, assuming all terms are non-zero so that LM is defined, this only works if $s=1$: taking the sum of coefficients on both sides of the equation you obtain the equation $1=s$. So you can never exchange a non-trivial sum and LM.
Note that only the image of LM needs to be of characteristic 0: this still holds for any $k$ if you view LM as a map from $k[x_1,\dots,x_n]$ to $\mathbb Z[x_1,\dots,x_n]$.
