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On Hilbert's 17th problem and Pfister's multiplicative formulae for the ring of real analytic functions

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2014
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Acquistapace, Francesca
Broglia, Fabrizio
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SCUOLA NORMALE SUPERIORE DI PISA
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In this work, we present "infinite" multiplicative formulae for countable collections of sums of squares (of meromorphic functions on R-n). Our formulae generalize the classical Pfister's ones concerning the representation as a sum of 2(r) squares of the product of two elements of a field K which are sums of 2(r) squares. As a main application, we reduce the representation of a positive semidefinite analytic function on R-n as a sum of squares to the representation as sums of squares of its special factors. Recall that roughly speaking a special factor is an analytic function on R-n which has just one complex irreducible factor and whose zeroset has dimension between 1 and n - 2.
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