I know that trivially the OR function on $n$ variables $x_1,\ldots, x_n$ can be represented exactly by the polynomial $p(x_1,\ldots,x_n)$ as such:$p(x_1,\ldots,x_n) = 1-\prod_{i = 1}^n\left(1-x_i\right)$,which is of degree $n$.But how could I show, what seems obvious, that if $p$ is a polynomial that represents the OR function exactly (so $\forall x \in \{0,1\}^n : p(x) = \bigvee_{i = 1}^n x_i$), then $\deg(p) \ge n$?
Representing OR with polynomials
cc.complexity theory;co.combinatorics;lower bounds;polynomials