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I'm self-learning about Model Theory and I just got to the proof of Hilbert's 17th Problem via Model Theory of Real Closed Fields. The 17th problem asks to show that a non-negative rational function must be the sum of squares of rational functions.

It seems to me that I lack a strong enough understanding of the context of the problem to understand its significance. Why did Hilbert think it was important enough to include it as one of his 23 problems?

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    $\begingroup$ Hilbert had done work on this topic already and knew a positive-definite polynomial in 2 variables need not be a sum of squares of polynomials (working over real numbers) but is a sum of squares of rational functions. So the question in $n$ variables interested him. To him, the problems he posed were not supposed to be anything like the most important questions for the 20th century. They were just “particular problems [from] which an advancement of science may be expected.” $\endgroup$ – KCd Jan 24 at 19:25
  • $\begingroup$ Continuing on with the "advancement of science" theme, there was a Quanta article from 2018 ("A Classical Math Problem Gets Pulled Into the Modern World") that describes how Hilbert's 17th problem plays a role in certain "real-world" optimization problems. No doubt far from Hilbert's motivation for inclusion on his list, but it shows that the problem had merit beyond simply satisfying an idle curiosity. $\endgroup$ – Mark Yasuda Jan 29 at 2:43

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