Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\}$$ i.e. as the number of non-zero terms.
By playing a bit with the Taylor series of $\sqrt{1-x}$ in $x=0$ it is not difficult to prove that for an infinite number of rational polynomials $W(p^2)=W(p)$ holds. Moreover, it is well-known that for an infinite number of rational polynomials, with degree at least $12$,
$$W(p^2) = W(p)-1$$ holds. Now I wonder:
• Is there some $p\in\mathbb{Q}[x]$ such that $W(p^2)=W(p)-2$?
• Is $W(p)-W(p^2)$ bounded by some constant?
• If not, is it possible to solve $W(p)-W(p^2)\geq k$ through some tensor trick?
• What is the minimum degree of a polynomial for which $W(p)-W(p^2)=k$?