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

. 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 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