The set of all polynomial functions from $\mathbb{Z}^3 \rightarrow \mathbb{Z}/(2)$

Question

Let $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}_2$ be a polynomial function in $\mathbb{Z}[x_1, x_2, x_3]$. Then $f$ has the form $f(x_1, x_2, x_3) = c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_1 x_2 + c_5 x_1 x_3 + c_6 x_2 x_3 + c_7 x_1 x_2 x_3$ where $c_1, \dots ,c_7$ are some constants in $\mathbb{Z}_2$. Is that true?

Answer 1

Yes.

The interesting fact here is that $x^2 \equiv x \pmod 2$, so there is no reason to write down higher powers of $x$. So instead, you have all the possible combinations of the $x_i$ with some coefficient.