I'm looking at the proof of this lemma:Lemma For every integer $t>0$, there exists a (proper) polynomial of total degree $(2t)^d$ that differs with $C$ on at most $size(C) 2^{n-t}$ inputsWhere C is a d-depth circuit with OR and NOT gates, and the polynomial is on the field $\mathbb{F}_3=\{-1,0-1\}$.The construction of the polynomial is inductive:for all inputs $x_i$ we associate to it the variable $x_i$suppose a gate has input the polynomials $p_1,...,p_k$, let choose random subsets $S_1,...,S_t\subseteq [k]$, and define $q_i=(\sum_{j\in S_i}p_j)^2$, then the output polynomial is defined as $p=1-\prod_{i=1}^t (1-q_i)$A polynomial is said to be proper if given inputs in $\{0,1\}$ the output is in $\{0,1\}$. What I don't understand is the observation that if all the $p_i$'s are proper that so is $p$.
Approximating circuits with polynomial of low degree, can't understand small detail in the proof
circuit complexity;polynomials