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

• Francesca Acquistapace ^[1] ; Fabrizio Broglia ^[1] ; José F. Fernando ^[2]
  1. [1] University of Pisa

     University of Pisa

     Pisa, Italia

     
  2. [2] Universidad Complutense de Madrid

     Universidad Complutense de Madrid

     Madrid, España

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 2, 2014, págs. 333-369
• Idioma: inglés
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• Resumen

  • In this work, we present “infinite” multiplicative formulae for countable collections of sums of squares (of meromorphic functions on Rn). Our formulae generalize the classical Pfister’s ones concerning the representation as a sum of 2r squares of the product of two elements of a field K which are sums of 2r squares. As a main application, we reduce the representation of a positive semidefinite analytic function on Rn 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 Rn which has just one complex irreducible factor and whose zeroset has dimension between 1 and n − 2.