Resumen:
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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 ??Âr of F=0, and for every integer ??0, there exists a solution y?Ar of F=0 such that y?? mod m? 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 ? of A and the completion  of A are real curve germs and p(A)?p(Â), p(?)=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 ?: Every real AP-curve with value semigroup ? is non-Pythagorean if and only if there are q,p1,p2 ? ? with q
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