f:{0,1}n→{0,1}PQdegQ≤degPxkik≥2xi. So we can restrict our attention to multilinear polynomials.
Claim: The polynomials {∏i∈Sxi:S⊆[n]}, as functions {0,1}n→R form a basis of for the space of all functions {0,1}n→R.
Proof: We first show that the polynomials are linearly independent. Suppose that f=∑ScS∏i∈Sxi=0 for all (x1,…,xn)∈{0,1}n. We prove by (strong) induction on |S| that cS=0. Suppose that cT=0 for all |T|<k, and let us be given a set S of cardinality k. For all T⊂S we know by induction that cT=0, and so 0=f(1S)=cS, where 1S is the input which is 1 on the coordinates of S. □
The claim shows that the multilinear representation of a function f:{0,1}n→{0,1} is unique (indeed, f doesn't even have to be 0/1-valued). The unique multilinear representation of OR is 1−∏i(1−xi), which has degree n.