affine vector space

Question

In an article it is written that:
"A well known result from the theory of Boolean functions is that if the algebraic degree of a Boolean functions is less than d, then the sum over the outputs of the function applied to all elements of an affine vector space of dimension $\geq$d is zero."
Could anyone explains quote? If possible with an example.
Thanks in advance.

Answer 1

Suppose a Boolean function $f$ is represented as a polynomial of degree $< d$ in indeterminates $x_1, \ldots, x_m$ over the binary field $\mathbb F_2$, where $m \ge d$. Each term must involve fewer than $d$ indeterminates, so there is at least one indeterminate it does not contain. Thus

$$ \sum_{(x_1, \ldots, x_m) \in \mathbb F_2^m} f(x_1,\ldots,x_m) = 0 $$ because the summation of a term over an indeterminate not present is $0$ (i.e. $\sum_{x \in \mathbb F_2} t = t+ t = 0$).