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Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients such that $$\left(\sum\limits_{k=0}^{20} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$ I asked this question here, but I received no answer.

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This is a simple linear programming problem, which has no solutions.

Here are the calculations, in Mathematica:


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  • $\begingroup$ Can you say if $(\sum \limits_{k=0}^7 x^k) \times (\sum\limits_{k=0}^{8} x^k)=(2+x+x^2)P(x)+(1+x+2x^2)Q(x)$ have no solutions too ? $\endgroup$
    – Dattier
    Apr 14 at 21:43
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    $\begingroup$ @Dattier : I think such a question can be addressed similarly. $\endgroup$ Apr 14 at 21:50
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    $\begingroup$ @Dattier : The identity in your comment holds if e.g. $4096Q(x)=3 x^{12} (535 + 501 x)$ and $4096P(x)=1090 x^{13}+2389 x^{12}+4611 x^{11}+5390 x^{10}+5868 x^9+7928 x^8+9008 x^7+7904x^6+6848 x^5+6016 x^4+4864 x^3+3584 x^2+3072 x+2048$. $\endgroup$ Apr 14 at 21:57
  • $\begingroup$ And if $Q$ and $P$ have for coefficients natural numbers ? $\endgroup$
    – Dattier
    Apr 14 at 22:01
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    $\begingroup$ $P(x)=x^3 \left(x^{10}+x^9+3 x^5+x^3+2 x+1\right)$ and $Q(x)=x^{10}+2 x^9+x^7+3 x^5+x+1$ will do (if this was the last question on this page :-)). $\endgroup$ Apr 15 at 1:06

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