I'm referring to the proof by Shamir.
The polynomials transmitted in the protocol are of degree $\leq n$. Why does it mean that the polynomials could be transmitted in a polynomial size message?
Is there some bound on the size of the coefficients? Are they all representable (in binary representation) in polynomial space, and if so, why?
For the proof that $\textsf{PSPACE} \subseteq \textsf{IP}$, we may reduce from $\textsf{TQBF}$ when the Boolean formulas are in 3CNF. Suppose our formula has $m$ clauses and $n$ quantifiers. Once we arithmetize the 3CNF formula (before handling the quantifiers), we have a polynomial of degree $3m$ with maximum coefficient $m$. Now if we arithmetize the quantifiers without reducing the coefficients, we have a polynomial of degree at most $2^{n}3m$. After applying the reduction operators, each coefficient is at most $O(2^{n}3m)$. So a single coefficient can be written down in polynomial-space using $O(n + \log(m))$ bits.
The real concern is whether our polynomial has degree $\text{poly}(m, n)$. Certainly, a degree $2^{n}3m$ polynomial can require more than polynomial space. Hence, the need for the reduction operators. The coefficients themselves are not of concern though, even with the reduction operators.