Note that the inversion formula lets us expand the domain of from to ,
as a multilinear (actually multiaffine) function.
The degree of is then the same as the degree of this multilinear polynomial.
Lemma
For every there is a unique multilinear such that .
Proof
Existence is clear by the inversion formula.
Now suppose is multilinear and vanishes on .
Show this by induction on .
If the result is clear.
Now assume the result for and let vanish on .
Then for all we can write
Also so is multilinear.
Thus write
Let .
Then clearly so by induction hypothesis .
Now .
Setting , we find that .