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Some concrete aspects of Hilbert’s 17th problem. (English) Zbl 0972.11021

Delzell, Charles N. (ed.) et al., Real algebraic geometry and ordered structures. AMS special session, Baton Rouge, LA, USA, April 17-21, 1996 and the associated special semester at Louisiana State University and Southern University, Baton Rouge, LA, USA, January-May 1996. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 253, 251-272 (2000).
This paper is an excellent survey on representations as sum of squares of rational functions of positive semidefinite polynomials which are not a sum of squares of polynomials.
The paper starts with a lively historical presentation of Hilbert’s 17th Problem and classical results. Then it includes recent developments of this question as Choi-Lam, Lax-Lax and Schmüdgen examples.
Turning to the shape of denominators in the decomposition, the paper presents Polya’s and Habicht’s works, and a major result of the author (1995) on the existence – for each positive definite form – of a decomposition as sum of squares of rational functions with uniform denominators. Following Becker’s work, extension to sums of \(2k\)th powers of rational functions is also studied. The paper ends with the author’s amazing explicit decomposition which writes \(1+t^2/ 2+t^2\) as a sum of \(2k\)th powers in \(\mathbb{R}(t)\).
For the entire collection see [Zbl 0936.00030].

MSC:

11E10 Forms over real fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-03 History of number theory
11E76 Forms of degree higher than two
01A60 History of mathematics in the 20th century
14P99 Real algebraic and real-analytic geometry
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