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On the pythagoras numbers of real analytic set germs.

Fernando Galván, José Francisco and Ruiz Sancho, Jesús María (2005) On the pythagoras numbers of real analytic set germs. Bulletin de la Société Mathématique de France , 133 (3). pp. 349-362. ISSN 0037-9484

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Abstract

We Show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

Item Type:Article
Uncontrolled Keywords:Pythagoras number; sum of squares; M. Artin’s approximation.
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:15159
References:

Abhyankar (S.S.) – Resolution of singularities of embedded algebraic surfaces, 2nd, enlarged ed., Springer Monographs in Math., Springer Ver- lag, Berlin-Heidelberg-NewYork, 1998.

Andradas (C.), Br¨ocker (L.) & Ruiz (J.M.) – Constructible Sets in Real Geometry, Ergeb. Math. Grenzgeb., vol. 33, Springer Verlag, Berlin- Heidelberg-NewYork, 1996.

Becker (J.) & Gurjar (R.) – Curves with large tangent space, Trans. Amer. Math. Soc., t. 242 (1975), pp. 285–296.

Bochnak (J.), Coste (M.) & Roy (M.-F.) – Real Algebraic Geome- try, Ergeb. Math. Grenzgeb., vol. 36, Springer Verlag, Berlin-Heidelberg- New York, 1998.

Bourbaki (N.) – Commutative Algebra, Hermann, Paris, 1972.

Campillo (A.) & Ruiz (J.M.) – Some Remarks on Pythagorean Real Curve Germs, J. Algebra, t. 128 (1990), pp. 271–275.

Choi (M.D.), Dai (Z.D.), Lam (T.Y.) & Reznick (B.) – The Pythago- ras number of some affine algebras and local algebras, J. reine angew. Math., t. 336 (1982), pp. 45–82.

Fernando (J.F.) – On the Pythagoras numbers of real analytic rings, J. Algebra, t. 243 (2001), pp. 321–338.

Sums of squares in real analytic rings, Trans. Amer. Math. Soc., t. 354 (2002), pp. 1909–1919.

Analytic surface germs with minimal Pythagoras number, Math. Z., t. 244 (2003), pp. 725–752.

Fernando (J.F.) & Quarez (R.) – Some remarks on the computation of Pythagoras numbers of real irreducible algebroid curves through Gram matrices, J. Algebra, t. 274 (2004), pp. 64–67.

Fernando (J.F.) & Ruiz (J.M.) – Positive semidefinite germs on the cone, Pacific J. Math., t. 205 (2002), pp. 109–118.

Hironaka (H.) – Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., t. 79 (1964), pp. 109–123, 205–326.

de Jong (T.) & Pfister (G.) – Local Analytic Geometry, basic the- ory and applications, Advanced Lectures in Mathematics, Vieweg Verlag, Braunschweig-Wiesbaden, 2000.

Kurke (H.), Mostowski (T.), Pfister (G.), Popescu (D.) & Roczen (M.) – Die Approximationseigenschaft lokaler Ringe, Lect. Notes in Math., vol. 634, Springer Verlag, 1978.

Merrien (J.) – Un th´eor`eme des z´eros pour les id´eaux de s´eries formelles `a coefficients r´eels, C. R. Acad. Sci. Paris S´er. A-B, t. 276 (1973), pp. 1055– 1058.

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

Quarez (R.) – Pythagoras numbers of real algebroid curves and Gram matrices, J. Algebra, t. 238 (2001), pp. 139–158.

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