A multilinear polynomial is a polynomial whose monomials have exponent at
most 1 in each variable. It may have arbitrary total
degree, bounded by the number of variables. For
example, 
is a multilinear polynomial, whereas 
is not. The word multilinear here allows a constant term, so it does not mean a multilinear
form.
Every function 
has a unique multilinear polynomial agreeing
with it on this finite set. It
is supplied by its Boolean Fourier expansion
(O'Donnell 2021, pp. 20-25). Powers can be reduced
using 
on this set, which preserves the values of a polynomial
there but need not preserve its total degree.
