Open questions concernings Hilbert's 17th Problem for analytic curves

Abstract
The Hilbert 17th Problem asks when a psd function is a sum of squares, and of how many.

For real analytic curves this reduces to the local problem
for germs at singular points. For those germs, the problem splits into the
consideration of their irreducible branches.

Now, irreducible curve germs are classically discussed
using the semigroup of values: all irreducible curves with fixed semigroup
form a "moduli" algebraic set in some finite dimensional affine space.

There, Pythagoras numbers, positive semidefinite germs, sum of squares
provide semialgebraic mappings on and stratifications of the "moduli" set.

The understanding od these semialgebraic data is a difficult matter that
has surprising connections with classical concepts (for instance, Arf
curves and Pythagorean curves are one and the same thing).