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 $f=\sum_{\alpha}a_{\alpha}x^{\alpha}.$