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). Powers can be reduced using 
on this set, which preserves
the values of a polynomial there but need not preserve
its polynomial degree.
