On pythagorean real irreducible algebroid curves

Abstract

In this note we deal with the pythagoras number p of certain 1-dimensional rings, i.e., real irreducible algebroid curves over a real closed ground field k. The problem we are concerned with is to characterize those real irreducible algebroid curves which are pythagorean (i.e., p = 1). We obtain two theorems involving the value-semigroup. Then we apply them to solve the cases of: (a) Gorenstein curves, (b) planar curves, (c) monomial curves, and (d) curves of multiplicity <= 5. Finally, two conjectures are stated.