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Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if the rational functions could have coefficients over Q rather than over R. Here is the relavant part of his speech

"At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented."

Does anyone know what these "certain geometrical constructions" are?

It seems maybe to me that Hilbert was attempting to embed rational projective space into higher dimensional rational projective space via these polynomials. Briefly, given a nonnegative homogeneous function $f(x_0,\ldots, x_n)$ with rational coefficients induces a metric on $QP^n$. Suppose Hilbert's dream holds that $f=p_0^2+\cdots + p_N^2$ where $p_i$'s are polynomials with rational coefficients. Then the map $p: QP^n\to QP^N$ where $p(x)=(p_0(x),\ldots, p_N(x))$ is an isometric embedding (almost!) where the metric induced by $f$ is the pullback back of the Euclidean metric on $QP^N$.

The above is just my hazard. But I would be delighted if anyone is aware of what exactly Hilbert's intended "geometrical constructions" are.

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up vote 8 down vote accepted

Actually the answer is in the sections 36 to 39 of Hilbert's "Foundations of geometry", which can be found on the web. The constructions are construction with "straightedge" (ruler) and "transferrer of segments". I quote a result from Hilbert's book :

Theorem 41. A problem in geometrical construction is, then, possible of solution by the drawing of straight lines and the laying off of segments, that is to say, by the use of the straight-edge and a transferrer of segments, when and only when, by the analytical solution of the problem, the co-ordinates of the desired points are such functions of the co-ordinates of the given points as may be determined by the rational operations and, in addition, the extraction of the square root of the sum of two squares.

This result explains relatively clearly why this kind of geometrical constructions leads to the question of the determination of those functions of $x_1,\ldots,x_n$ which can be written as sums of squares of rational functions with rational coefficients.

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