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The Pythagoras number of a ring A is the smallest integer p A = p ≥ 1 such that any sum of squares of A is a sum of p squares, and p A = +∞ if such an integer does not exist. This is a very delicate… (More)

- José F. Fernando
- 2002

Let A be an analytic ring. We show: (1) A has finite Pythagoras number if and only if its real dimension is ≤ 2, and (2) if every positive semidefinite element of A is a sum of squares, then A is… (More)

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesus Ruiz
- 2005

We show that the Pythagoras number of a real analytic curve is the supremum of the Pythagoras numbers of its singularities, or that supremum plus 1. This includes cases when the Pythagoras number is… (More)

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesus 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… (More)

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f(x1, . . . , xn) with coefficients in the polynomial ring k[t] is a sum of 2n · τ(k) squares of linear forms,… (More)

Let K ⊂ R be a convex polyhedron of dimension n. Denote S := R \K and let S be its closure. We prove that for n = 3 the semialgebraic sets S and S are polynomial images of R . The former techniques… (More)

- Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesus 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… (More)

The problem of representing a positive semidefinite function (=psd) as a sum of squares (=sos) is a very old matter in real algebra and real geometry. Still, it is a difficult question always… (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 and have the property that every positive semidefinite analytic function germ is… (More)

- Francesca Acquistapace, Jesus Ruiz, Fabrizio Broglia, José F. Fernando
- 2005

— We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic… (More)