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Vladikavkaz. Mat. Zh., 2015, Volume 17, Number 2, Pages 32–36 (Mi vmj541)  

Artin's theorem for $f$-rings

A. G. Kusraev

Southern Mathematical Institute, Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia

Abstract: 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.

Key words: $f$-ring, complete ring of quotients, real closure, polynomial, rational function, Artin's theorem, Hilbert 17th problem, Boolean valued representation.

Full text: PDF file (202 kB)
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UDC: 510.67+512.55
MSC: 03C25, 12D15, 13B25
Received: 16.02.2015
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Citation: A. G. Kusraev, “Artin's theorem for $f$-rings”, Vladikavkaz. Mat. Zh., 17:2 (2015), 32–36

Citation in format AMSBIB
\Bibitem{Kus15}
\by A.~G.~Kusraev
\paper Artin's theorem for $f$-rings
\jour Vladikavkaz. Mat. Zh.
\yr 2015
\vol 17
\issue 2
\pages 32--36
\mathnet{http://mi.mathnet.ru/vmj541}


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