Sums of squares in planar Nash rings

Sums of Squares in Real Analytic Rings

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 real and has real dimension 2.

Sums of Squares in Real Rings

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.

On Sums of Squares in Local Rings

Let A be a semilocal ring. We compare the set of positive semidefinite (psd) elements of A and the set of sums of squares in A. For psd f ∈ A, whether f is a sum of squares or not depends only on the behavior of f in an infinitesimal neighborhood of the real zeros of f in Spec A. We apply this observation, first to 1-dimensional local rings, then to 2-dimensional regular semilocal rings. For th...

Sums of squares in excellent henselian local rings

This is a survey about the representation of positive semidefinite elements as sums of squares from a local viewpoint, in the sense of analytic geometry. Here, we present the most recent results in the more general frame of excellent henselian local rings with real closed residue field. 1

Subdirect Sums of Rings