Complutense University Library

On the Pythagoras numbers of real analytic rings

Fernando Galván, José Francisco (2001) On the Pythagoras numbers of real analytic rings. Journal of Algebra, 243 (1). pp. 321-338. ISSN 0021-8693

[img] PDF
Restricted to Repository staff only until 2020.

158kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0021869301988696

View download statistics for this eprint

==>>> Export to other formats

Abstract

We show that the Pythagoras number of a real analytic ring of dimension 2 is finite, bounded by a function of the multiplicity and the codimension.

Item Type:Article
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:15239
References:

C. Andradas, L. Br¨ocker, and J. M. Ruiz, “Constructible Sets in Real Geometry,”Ergeb. Math., Vol. 33, Springer-Verlag, Berlin/Heidelberg/New York, 1996.

M. Artin, On the solution of analytic equations, Invent. Math. 5 (1968), 227–291.

J. Bochnak, M. Coste, and M. F. Roy, “Real Algebraic Geometry,” Ergeb. Math.,Vol. 36, Springer-Verlag, Berlin/Heidelberg/New York, 1998.

J. Bochnak, W. Kucharz, and M. Shiota, On equivalence of ideals of real global analytic functions and the 17th Hilbert problem, Invent. Math. 63 (1981), 403–421.

J. Bochnak and J.-J. Risler, Le th´eor`eme des z´eros pour les vari´et’es analytiques r´eelles de dimension 2, Ann. Sci. Ecole Norm. Sup. 4 8 (1975), 353–364.

J. W. S. Cassels, W. S. Ellison, and A. Pfister, On sums of squares and on elliptic curves over functions fields, J. Number Theory 3 (1971), 125–149.

A. Campillo and J. M. Ruiz, Some remarks on Pythagorean real curve germs,J. Algebra 128 (1990), 271–275.

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. 336 (1982), 45–82.

M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials, in Proc. Sympos. Pure. Math. Vol. 58, 103–126, Amer. Math. Soc., Providence, 1995.

D. Z. Djokovi´c, Hermitian matrix over polynomial rings, J. Algebra 43 (1976),359–374.

Ch. N. Diller and A. Dress, Zur Galoistheorie pythagor¨aischer K¨orper, Arch. Math.16 (1965), 148–152.

J. F. Fernando, Positive semidefinite germs in real analytic surfaces, Mathematische Annalen, in press.

J. F. Fernando, “Sumas de cuadrados en g´ermenes de superficie” Tesis doctoral,Universidad Complutense, Madrid, 2001.

J. F. Fernando and J. M. Ruiz, Positive semidefinite germs on the cone, Pacific J.Math. in press.[Hu] T. W. Hungerford, “Algebra,” Grad. Text in Math., Vol. 73, Springer-Verlag,Berlin/New York, 1974.

P. Jaworski, About estimates on number of squares necessary to represent a positive-semidefinite analytic function, Arch. Math. 58 (1992), 276–279.

T. de Jong and G. Pfister, “Local Analytic Geometry, Basic Theory and Applications,”Advanced Lectures in Mathematics, Vieweg, Braunschweig/Wiesbaden,2000.

E. Kunz, “Introduction to Commutative Algebra and Algebraic Geometry,”Birkh¨auser,Boston/Basel/Stuttgart,1985.

H. Kurke, T. Mostowski, G. Pfister, D. Popescu, and M. Roczen, “Die Approximationseigenschaft lokaler Ringe,” Lecture Notes in Math., Vol. 634, Springer-Verlag,Berlin, 1978.

S. Lang, “Algebra,” Addison–Wesley, Reading, MA, 1965.

J. Ortega, On the Pythagoras number of a real irreducible algebroid curve, Math.Ann. 289 (1991), 111–123.

A. Pfister, Zur Darstellung definiter Funktionen als Summe von Quadraten, Invent.Math. 4 (1967), 229–237.

R. Quarez, Pythagoras numbers of real algebroid curves and Gram matrices,preprint, Univ. Rennes I, 1998.

J. M. Ruiz, On Hilbert’s 17thproblem and real nullstellensatz for global analytic functions, Math. Z. 190 (1985), 447–449.

J. M. Ruiz, “The Basic Theory of Power Series,” Advanced Lectures in Mathematics,Vieweg,Braunschweig/Wiesbaden, 1993.

J. M. Ruiz, Sums of two squares in analytic rings, Math. Z. 230 (1999), 317–328.

C. Scheiderer, On sums of squares in local rings, preprint, Univ. Duisburg, 2000. J. M. Ruiz, “The Basic Theory of Power Series,” Advanced Lectures in mathematics,Vieweg,Braunschweig/Wiesbaden, 1993.

J. M. Ruiz, Sums of two squares in analytic rings, Math. Z. 230 (1999), 317–328.

C. Scheiderer, On sums of squares in local rings, preprint, Univ. Duisburg, 2000.

Deposited On:17 May 2012 08:21
Last Modified:06 Feb 2014 10:19

Repository Staff Only: item control page