Sur certaines algèbres de fonctions analytiques. (On some algebras of analytic functions). (French) Zbl 0634.14017

Tougeron, Jean-Claude

Sémin. géométrie algébrique réelle, Paris 1986, tome 1, Publ. Math. Univ. Paris VII 24, 35-121 (1986).

[For the entire collection see Zbl 0623.00004.]
In this important paper the author extends to some analytic sets the elementary properties of real algebraic properties: (i) a decreasing sequence of real algebraic varieties is stationary, (ii) Bezout theorem; (iii) Lojasiewicz inequality.
Some results of this kind have been obtained by Khovanski for Pfaffian varieties. The author’s construction is quite different from Khovanski’s and leads him to the definition of Liouville varieties. He considers subalgebras \({\mathcal O}(\Omega)\) of the algebra \({\mathcal H}(\Omega)\) of real analytic functions on an open \(\Omega\) of \({\mathbb{R}}^ n\) and studies the following type of properties:
(1) algebraic property: \(\Omega\) with the topology of the maximal spectrum of \({\mathcal O}(\Omega)\) is noetherian,
(2) topological property: if E is a finite diemensional vectorial subspace of \({\mathcal O}(\Omega)\) and if \(f_ 1,...,f_ n\in E\) are such that the zeros of \(f=(f_ 1,...,f_ n)\) are degenerate, then the number of zeroes of f is uniformally bounded by a \(N(E),\)
\((3_{\Gamma})\) metric property: let \(\Gamma\) be a family of real continuous functions on \(\Omega\) with some simple conditions (cf. § 2 in the paper) then for all \(f\in \Omega\) there are \(\gamma\in \Gamma\) and \(\alpha >0\) such that for all \(x\in \Omega:\gamma(x)\geq | f(x)| \geq \gamma (x)^{-1}d(x,f^{-1}(0))^{\alpha}.\)
Property (1) implies the existence of finite stratifications, and allows by a convenient notion of localization to extend local analytic geometry results to the generic point. Property (2) implies using Morse theory uniform bounds on Betti numbers. Properties \((1+3_{\Gamma})\) imply regular separation properties on semi-analytic sets and divisons theorem of some classes of \({\mathcal C}^{\infty}\) functions by \({\mathcal O}(\Omega)\) functions, so extending Hörmander-Lojasiewicz-Malgrange results. - The main results of the paper are the following:
A: (1) (resp. (2), resp. (1)\(+(3_{\Gamma}))\) for \({\mathcal O}(\Omega)\) implies (1) (resp. (2), resp. (1)\(+(3_{\Gamma [y]}))\) for \({\mathcal O}(\Omega)[y].\)
B: Let \(\phi\in \Omega\) be such that for \(i=1,...,n\), \(P(\phi)\cdot \phi '_{X_ i}=Q_ i(\phi)\) (P and \(Q_ i\) are polynomials with coefficients in \({\mathcal O}(\Omega)\) and P does not vanish identically on each connected component of \(\Omega\) ; then (1)+(2) for \({\mathcal O}(\Omega)\) implies \((1)+(2)\) for \({\mathcal O}(\Omega)[D^{\omega}\phi]\), \(\omega \in {\mathbb{N}}^ n.\)
C: \((1)+(2)+(3_{\Gamma})\) for \({\mathcal O}(\Omega)\) implies \((1)+(2)+(3_{\Gamma})\) for \({\mathcal O}(\Omega)[\phi]\) in the following cases:
- the derivatives of \(\phi\) are in \({\mathcal O}(\Omega)\) and \(\phi\) is dominated by \(\Gamma\);
- \(\phi =e^{\theta}\), with \(\theta\in {\mathcal O}(\Omega)\), \(\Gamma'=\Gamma [e^{\theta}];\)
- \(\phi\) is algebraic on \({\mathcal O}(\Omega)\) (with an auxiliary hypothesis).
This results permit to construct by iteration, starting with polynomials or local analytic functions, Liouville classes which are by the preceding results classes of varieties and analytic functions sharing the elementary properties of real algebraic varieties and polynomials. - In section 1 \(\Omega\)-analytic noetherian varieties are introduced and studied (property (1))). Section 2 is devoted to global Lojasiewicz inequalities (property (3)). Section 3 studies Liouville algebras property (2)). In section 4 the definition and properties of Liouville classes are given. In section 5, properties of (global) analytic and semi-analytic sets for Liouville classes are studied.

Reviewer: M.F.Roy

MSC:

14Pxx	Real algebraic and real-analytic geometry
46J10	Banach algebras of continuous functions, function algebras
32B20	Semi-analytic sets, subanalytic sets, and generalizations

Keywords:

Liouville varieties; semi-analytic sets

Citations:

Zbl 0623.00004