The Nullstellensatz for real coherent analytic surfaces. (English) Zbl 1215.14058

Broglia, Fabrizio; Pieroni, Federica

Rev. Mat. Iberoam. 25, No. 2, 781-798 (2009).

Let \(X\) be a real coherent analytic space, \({\mathcal O}_X\) the sheaf of analytic functions, \({\mathcal O}(X)\) the ring of global analytic functions. Given an ideal \({\mathfrak a}\subseteq{\mathcal O}(X)\), the zero set of \({\mathfrak a}\) is denoted by \({\mathcal Z}({\mathfrak a})\), the vanishing ideal of \({\mathcal Z}({\mathfrak a})\) is denoted by \({\mathcal I}({\mathcal Z}({\mathfrak a}))\). Nullstellensätze describe the connections between the ideals \({\mathfrak a}\) and \({\mathcal I}({\mathcal Z}({\mathfrak a}))\): When is it true that \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\)?
The authors show that, given a real coherent analytic surface \(X\), the equality \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\) holds if and only if \({\mathfrak a}\) is a real ideal and is saturated (i.e., \({\mathfrak a}\) is the ideal of global sections of the ideal sheaf generated by \({\mathfrak a}\)). It remains an open question whether the result can be extended to the space \(\mathbb{R}^3\). However, it is shown that the result fails for \(\mathbb{R}^3\) if and only if there is a so-called special irreducible functions that generates a real ideal. Primary ideals and primary decompositions of saturated ideals are among the main tools of the paper.

Reviewer: Niels Schwartz (Passau)

MSC:

14P15	Real-analytic and semi-analytic sets
14P99	Real algebraic and real-analytic geometry
32B10	Germs of analytic sets, local parametrization
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

real analytic space; analytic function; zero set; vanishing ideal; Nullstellensatz; primary decomposition

References:

[1]	Acquistapace, F., Broglia, F., Fernando, J.F. and Ruiz, J.M.: On the finiteness of Pythagoras numbers of real meromorphic functions. Article · Zbl 1197.14060
[2]	Acquistapace, F., Broglia, F., Fernando, J.F.: On a global analytic Positivstellensatz. Ark. Mat. 47 (2009), no. 1, 13-39. · Zbl 1216.14053
[3]	Acquistapace, F., Broglia, F., Tognoli, A.: Sulla normalizzazione degli spazi analitici reali. Boll. Un. Mat. Ital. (4) 12 (1975), no. 1-2, 26-36. · Zbl 0318.32001
[4]	Adkins, W.A.: A real analytic Nullstellensatz for two dimensional manifolds. Boll. Un. Mat. Ital. B (5) 14 (1977), no. 3, 888-903. · Zbl 0444.14019
[5]	Adkins, W.A. and Leahy, J.V.: A global real analytic Nullstellensatz. Duke Math. Journal 43 (1976), no. 1, 81-86. · Zbl 0353.32009
[6]	Andradas, C., Bröcker, L. and Ruiz, J.M.: Constructible sets in real geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 33 . Springer-Verlag, Berlin, 1996. · Zbl 0873.14044
[7]	Andradas, C. and Díaz-Cano, A.: Some properties of global semianalytic subsets of coherent surfaces. Illinois J. Math. 48 (2004), no. 2, 519-537. · Zbl 1077.14087
[8]	Atiyah, M.F. and Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601
[9]	Bochnak, J., Coste, M. and Roy, M.F.: Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 36 . Springer-Verlag, Berlin, 1998. · Zbl 0912.14023
[10]	Bochnak, J. and Risler, J.J.: Le théorème des zéros pour les variétés analytiques réelles de dimension \(2\). Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 353-363. · Zbl 0315.14008
[11]	Cain, B.E.: A two color theorem for analytic maps in \(\R ^n\). Proc. Amer. Math. Soc. 39 (1973), 261-266. · Zbl 0236.26012
[12]	Coen, S.: Sul rango dei fasci coerenti. Boll. Un. Mat. Ital. (3) 22 (1967), 373-382. · Zbl 0164.38202
[13]	De Bartolomeis, P.: Algebre di Stein nel caso reale. Rend. Accad. Naz. XL (5) 1/2 (1975/76), 105-144 (1977). · Zbl 0353.32010
[14]	De Bartolomeis, P.: Una nota sulla topologia delle algebre reali coerenti. Boll. Un. Mat. Ital. (5) 13A (1976), no. 1, 123-125. · Zbl 0353.32011
[15]	Fernando, J.F.: On the 17th Hilbert Problem for global analytic functions on dimension \(3\). To appear in Commentarii Math. Helvetici .
[16]	Forster, O.: Primärzerlung in Steinschen Algebren. Math. Annalen 154 (1964), 307-329. · Zbl 0135.12602
[17]	Galbiati, M.: Stratifications et ensemble de non-cohérence d’un espace analytique réel. Invent. Math. 34 (1976), no. 2, 113-128. · Zbl 0314.32006
[18]	Galbiati, M. and Tognoli, A.: Alcune proprietà delle varità algebriche reali. Ann. Scuola Norm. Sup. Pisa (3) 27 , 359-404 (1973). · Zbl 0277.14010
[19]	Pieroni, F.: A sufficient condition for a real global Nulstellensatz.
[20]	Ruiz, J.: On Hilbert’s 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190 (1985), no. 3, 447-454. · Zbl 0579.14021
[21]	Ruiz, J.: The basic theory of power series. Advanced Lectures in Mathematicas. Friedr. Vieweg & Sohn, Braunschweig, 1993.
[22]	Siu, Y.T.: Hilbert Nullstellensatz in global complex analytic case. Proc. Amer. Math. Soc. 19 (1968), 296-298. JSTOR: · Zbl 0162.10401
[23]	Tognoli, A.: Proprietà globali degli spazi analitici reali. Ann. Mat. Pura Appl. (4) 75 (1967), 143-218. · Zbl 0158.33001