Publication: Pythagorean real curve germs
Loading...
Full text at PDC
Publication Date
1985-05
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Academic Press
Citations
Exportar
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.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M. ARTIN, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291.
S. BASARAB, V. NICA, AND D. POPESCU, Approximation properties and existencial completeness for ring morphisms, Manuscripta Math. 33 (1981), 227-282.
A. CAMPILLO, “Algebroid Curves in Positive Characteristic,” Lecture Notes in Mathematics No. 813, Springer-Verlag, New York, 1980.
M. D. CHOI, Z. D. DAI, T. Y. LAM, AND B. REZNICK, The pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math. 306 (1982), 45-82.
E. KUNZ, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970) 748-751.
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, Sobre álgebras de Nash. Actes VII JMHL, Pub. Mat. Univ. Autónoma de Barcelona, 20, 1980.
J. M. RUIZ, “Aspectos aritméticos y geométricos del problema decimoséptimo de Hilbert para gérmenes analíticos,” Ph.D. dissertation, Univ. Complutense de Madrid, September, 1982.
0. ZARISKI AND P. SAMUEL,“Commutative Algebra I,” Graduate Texts in Mathematics No. 28, Springer-Verlag, New York, 1979.