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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 $W(p)-W(p^2)=k$?
    Crude bounds are also welcome.
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