Artin's theorem for $f$-rings
A. G. Kusraev
Southern Mathematical Institute, Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia
The main result states that each positive polynomial $p$ in $N$ variables with coefficients in a unital Archimedean $f$-ring $K$ is representable as a sum of squares of rational functions over the complete ring of quotients of $K$ provided that $p$ is positive on the real closure of $K$. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.
$f$-ring, complete ring of quotients, real closure, polynomial, rational function, Artin's theorem, Hilbert 17th problem, Boolean valued representation.
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MSC: 03C25, 12D15, 13B25
A. G. Kusraev, “Artin's theorem for $f$-rings”, Vladikavkaz. Mat. Zh., 17:2 (2015), 32–36
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\paper Artin's theorem for $f$-rings
\jour Vladikavkaz. Mat. Zh.
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