Some remarks on pythagorean real curve germs

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Campillo, Antonio
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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.
M. ARTIN, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sri. Publ. Math. 36 (1969), 23-58. E. BECKER, The real holomorphy ring and sums of 2nth powers, in “Géométrie Algébrique Réelle et Formes Quadratiques,” Vol. 959, Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1982. A. CAMPILLO, “Algebroid Curves in Positive Characteristic,” Vol. 959, Lecture Notes in Mathematics, Springer-Verlag, New York, 1980. J. LIPMAN, Stable ideals and Arf rings, Amer. J. Math. 93 (1971) 649-685. M. NAGATA, “Local Rings,” Interscience, New York, 1962. J. M. RUIZ, Pythagorean real curve germs, J. Algebra 94 (1985), 126-144.