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?