Open AccessOn Hilbert's 17th problem and Pfister's multiplicative formulae for the ring of real analytic functions
Author(s) -
José F. Fernando,
Fabrizio Broglia,
José Francisco Fernando Galván
Publication year - 2014
Publication title -
annali della scuola normale superiore di pisa. classe di scienze
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.444
H-Index - 37
eISSN - 2036-2145
pISSN - 0391-173X
DOI - 10.2422/2036-2145.201201_004
Subject(s) - mathematics , multiplicative function , countable set , explained sum of squares , analytic function , pure mathematics , meromorphic function , ring (chemistry) , dimension (graph theory) , function (biology) , representation (politics) , field (mathematics) , combinatorics , mathematical analysis , statistics , evolutionary biology , politics , political science , law , biology , chemistry , organic chemistry
. 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