A linear transformation brings $\langle$takes$\rangle$ us back to the case in which ......
The solutions can be carried back to $H(V)$ with the aid of the mapping function $\phi$.
......(a result that dates back to a 1915 paper of Hadamard)
This idea goes back at least as far as [3].
This argument goes back to Banach.
Schenzel's formula frequently allows us to move back and forth between the commutative algebra of $k[P]$ and the combinatorics of $P$.
We now transfer the above analysis back to $M(A)$.
We now turn back to our main question.
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