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.
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.
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)
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
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)
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)
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.