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Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?

Euler's two squares factoring states that numbers of form $n=pq$ with $p,q$ of form $4k+1$ could be factored by gcd techniques if representations $n=a^2+b^2=c^2+d^2$ could be found. Is there an analog of this formulation over polynomial rings?

That is, I mean is there special form of polynomials $p(x)$, $q(x)$ such that $p(x)q(x)$ has two different sum of squares representation and special forms when such representations are not available?

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  • $\begingroup$ I doubt it, as most integer polynomials are irreducible; Try replacing $\mathbb{Z}$ with a finite field. $\endgroup$
    – Ofir Gorodetsky
    Feb 8, 2015 at 18:13
  • $\begingroup$ @OfirGorodetsky Technique exists over $\mathbb F_q[x]$? $\endgroup$
    – Turbo
    Jan 12, 2019 at 16:19

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