Some remarks on pythagorean real curve germs

Abstract

Let k be a real closed field. A real AP-curve (over k) is a 1-dimensional, excellent Henselian local real domain with residue field k. A 1-dimensional Noetherian local ring is Arf, if emb dim(B)=mult(B) for every local ring B infinitely near to A [ J. Lipman , Amer. J. Math. 93 (1971), 649–685]. For n≥1, the 2nth Pythagoras number p2n of a commutative ring A is the least p, 1≤p≤+∞, such that any sum of 2nth powers in A is a sum of no more than p2nth powers in A. A main purpose of this paper is to affirm the following conjectures proposed by Ruiz [J. Algebra 94 (1985), no. 1, 126–144]: Let A be a real AP-curve, and let A be Pythagorean (i.e., p2=1). Then (i) A is Arf. (ii) Every local ring infinitely near to A is Pythagorean. Actually, the authors obtain a finer result: For a real AP-curve A, the following assertions are equivalent: (1) A is Arf; (2) A is Pythagorean; (3) p2n=1 for some n; (4) p2n=1 for all n. Here, (2)(1) is exactly Conjecture (i) and (1)(2) reduces Conjecture (ii) to the obvious fact that, if A is Arf, every local ring infinitely near to A is Arf too. Of course, the result contains some additional insight into the study of Pythagoras's numbers, even of higher order, of real curve germs.