The Princeton Companion briefly mentions the general question was 'interesting' and 'difficult' without providing any reference. Can someone shed light on why this is so?
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1Single or multi-variable? Coefficients in...? – WimC Feb 12 '14 at 12:19
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Both. Why is the multivariate case particularly interesting? – user39914 Feb 12 '14 at 12:24
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The single-variable case is sort of trivial (at least over real). A real polynomial in single-variable can be written as a sum of two squares of real polynomials iff it is non-negative for all its argument. This isn't the case for polynomials over multi-variables. – achille hui Feb 12 '14 at 12:31
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For polynomials in a single variable the "Pythagorean triples" of polynomials are determined by more or less the same condition as Pythagorean triples of integers. So, if $p$, $q$, and $r$ are polynomials, then $p^2 + q^2 = r^2$ if and only if there exist polynomials $u$ and $v$ such that $p = u^2 - v^2$, $q = 2uv$, $r = u^2 + v^2$
See K. K. Kubota, "Pythagorean triples in unique factorization domains", The American Mathematical Monthly, Vol. 79, No. 5 (May, 1972), pp. 503-505.
This is interesting (to me) because it tells me how I can parameterize circles with rational functions. It's also the foundation of a class of curves called "Pythagorean hodograph" curves, which are useful because their offsets are easy to construct. See this answer, for example, or this one.
