Pythagorean real curve germs

Abstract

Let k be a real closed field. A real curve germ over k is a real one-dimensional Noetherian local integral domain with residual field k. A Noetherian local ring A with maximal ideal m and completion  is an AP-ring if for every system of polynomials F∈A[Y]s, Y=(Y1,⋯,Yr), for every formal solution ŷ∈Âr of F=0, and for every integer λ≥0, there exists a solution y∈Ar of F=0 such that y≡ŷ mod mλ Â. A real AP-curve is a real curve germ which is an AP-ring. The Pythagoras number p(A) of A is the least p, 1≤p≤+∞, such that each sum of squares in A is a sum of p squares. The author proves that for any real AP-curve A (over a real closed field) the derived normal ring Ā of A and the completion  of A are real curve germs and p(A)≤p(Â)<∞, p(Ā)=1. The value semigroup of a real AP-curve is a numerical semigroup, that is, an additive subsemigroup of the nonnegative integers, whose complement is finite. The main theorem classifies real AP-curves A which are Pythagorean (that is, p(A)=1) by their value semigroup Γ: Every real AP-curve with value semigroup Γ is non-Pythagorean if and only if there are q,p1,p2 ∈ Γ with q<p1≤p2 such that p1+p2−q∉Γ. Moreover, for a given numerical semigroup Γ the author proves: Every real AP-curve with value semigroup Γ is Pythagorean if and only if for each q∈Γ, p∈E with q<p, one has (q+c)/2≤p. Here c denotes the least positive integer such that Γ contains each p≥c and E is some specified subset of Γ. The paper ends with some applications: A Gorenstein real AP-curve is Pythagorean if and only if its multiplicity is ≤2. A monomial real AP-curve is Pythagorean if and only if it is Arf. There is a list of all Pythagorean real algebroid curves of multiplicity ≤5.