= Monomial containment in an ideal of coordinate powers
{title2=$(X_0^{d_0},\ldots,X_n^{d_n})\supseteq k[X_0,\ldots,X_n]_{\sum d_i}$}
For integers $d_i\ge0$, every <monomial> of total degree $\sum_i d_i$ is divisible by some $X_i^{d_i}$. If any $d_i=0$ the ideal is the whole ring. Otherwise a <monomial> avoiding all these divisibilities would have total degree at most $\sum_i(d_i-1)$, a contradiction. The sharper least degree containing every <monomial> is $\sum_i(d_i-1)+1$ when all $d_i>0$.
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