On pythagorean real irreducible algebroid curves (Q1064366)

A real irreducible algebroid curve A is a real complete local integral domain of dimension 1 with residue field a real closed field K. The Pythagoras number of A, pA, is the least p such that any sum of squares in A is a sum of p squares. A is called pythagorean if \(pA=1.\) In this note the author sketches the proof of two theorems characterizing pythagorean real algebroid curves in term of the valuation semi-group of their normalization. Several consequences are indicated, particularly results characterizing pythagorean real algebroid curves of low multiplicity.