Monomial containment in an ideal of coordinate powers ($(X_0^{d_0},\dots ,X_n^{d_n})\supseteq k[X_0,\dots ,X_n{]}_{\sum d_i}$)

For integers $d_i\ge 0$, 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 $d_i>0$.