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We show that any positive semidefinite analytic function germ on the cone z 2 = x 2 + y 2 is a sum of two squares of analytic function germs.

- JOSÉ F. FERNANDO
- 2004

Let A be an excellent ring. We show that if the real dimension of A is at least three then A has infinite Pythagoras number, and there exists a positive semidefinite element in A which is not a sum of squares in A.

- JOSÉ F. FERNANDO
- 2006

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f (x 1 ,. .. , xn) with coefficients in the polynomial ring k[t] is a sum of 2n · τ (k) squares of linear forms, where τ (k) is the supre-mum of the levels of the finite non-real field extensions of k. From this result we deduce bounds for the Pythagoras numbers of… (More)

We show that: (i) the Pythagoras number of a real analytic set germ is the supre-mum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythago-ras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples… (More)

- José F. Fernando
- 2002

We determine all complete intersection surface germs whose Pythagoras number is 2, and find they are all embedded in R 3 and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in R 3. Finally, we find in higher… (More)

Let R be a real closed field and n ≥ 2. We prove that: (1) for every finite subset

- José F. Fernando
- 2005

Among the invariant factors g of a positive semidefinite analytic function f on R 3 , those g whose zero set Y is a curve are called special. We show that if each special g is a sum of squares of global meromorphic functions on a neighbourhood of Y , then f is a sum of squares of global meromorphic functions. Here sums can be (convergent) infinite, but we… (More)

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz, Eberhard Becker
- 2004

We consider Hilbert's 17 problem for global analytic functions in a modified form that involves infinite sums of squares. This reveals an essential connection between the solution of the problem and the computation of Pythagoras numbers of meromorphic functions.

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz, Eberhard Becker
- 2005

We consider the 17th Hilbert Problem for global analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for an analytic function to be a sum of squares. We deduce that an affirmative solution to the 17th Hilbert Problem for global analytic functions implies the finiteness of the Pythagoras number… (More)

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz
- 2004

We show that (i) every positive semidefinite meromorphic function germ on a surface is a sum of 4 squares of meromorphic function germs, and that (ii) every positive semidefinite global meromorphic function on a normal surface is a sum of 5 squares of global meromorphic functions.