Geometry of subanalytic and semialgebraic sets: Abstract.

*(English)*Zbl 0870.32001
Broglia, Fabrizio (ed.) et al., Real analytic and algebraic geometry. Proceedings of the international conference, Trento, Italy, September 21-25, 1992. Berlin: Walter de Gruyter. 251-275 (1995).

The present paper explains the main results of the book “Geometry of subanalytic and semialgebraic sets” by M. Shiota, which the author is preparing, with sketches of proofs. The book intends to clarify where good properties of subanalytic and semialgebraic sets come from, by showing that any family of subsets of Euclidean spaces, which satisfies certain axioms, retains the properties:

Axiom (i) Every algebraic set in a Euclidean space is an element of \({\mathfrak X}\).

Axiom (ii) For elements \(X_1\subset \mathbb{R}^n\) and \(X_2\subset \mathbb{R}^n\) of \({\mathfrak X}\), \(X_1\cap X_2\), \(X_1-X_2\) and \(X_1\times X_2\) are elements of \({\mathfrak X}\).

Axiom (iii) Let \(X\subset \mathbb{R}^n\) be an element of \({\mathfrak X}\) and let \(p:\mathbb{R}^n\to \mathbb{R}^m\) be a linear map such that the restriction of \(p\) to \(\overline X\) is proper. Then \(p(X)\) is an element of \({\mathfrak X}\).

Axiom (iv) For an element \(X\subset \mathbb{R}\) of \({\mathfrak X}\), each point of \(X\) has a neighborhood in \(X\) which is a finite union of points and intervals. We introduce the axioms so that the families of all subanalytic sets and of all semialgebraic sets are typical examples of \({\mathfrak X}\)-families.

Examples of \({\mathfrak X} \)-families other than the families of subanalytic sets and of semi-algebraic sets are the following.

(a) The family of all locally semialgebraic sets.

(b) An 0-minimal Tarski system generated by the relations \(x+y=z\), \(x\cdot y=z\) and \(\exp (x)=y\). Note that the closure of the graph of the function \(y=\exp (-1/x^2)\) is an element of this system and \(C^\infty\) smooth but not \(C^\omega\) smooth.

(c) Let \(X\subset \mathbb{R}^n\) be a closed semialgebraic set, and let \(f_i\), \(i=1, \dots,k\), be Pfaffian functions on an open neighbourhood \(U\) of \(X\), i.e., for each \(i\) \({\partial f_i \over \partial x_j} (x)=P_j(x,f_i(x))\) on \(U\), \(j=1, \dots,n\), for some polynomial functions \(P_j\). Then each element of the smallest family \({\mathfrak X}\) containing graph \(f_i |_X\) is described by \(f_i\) and a finite number of polynomial functions.

For the entire collection see [Zbl 0812.00016].

Axiom (i) Every algebraic set in a Euclidean space is an element of \({\mathfrak X}\).

Axiom (ii) For elements \(X_1\subset \mathbb{R}^n\) and \(X_2\subset \mathbb{R}^n\) of \({\mathfrak X}\), \(X_1\cap X_2\), \(X_1-X_2\) and \(X_1\times X_2\) are elements of \({\mathfrak X}\).

Axiom (iii) Let \(X\subset \mathbb{R}^n\) be an element of \({\mathfrak X}\) and let \(p:\mathbb{R}^n\to \mathbb{R}^m\) be a linear map such that the restriction of \(p\) to \(\overline X\) is proper. Then \(p(X)\) is an element of \({\mathfrak X}\).

Axiom (iv) For an element \(X\subset \mathbb{R}\) of \({\mathfrak X}\), each point of \(X\) has a neighborhood in \(X\) which is a finite union of points and intervals. We introduce the axioms so that the families of all subanalytic sets and of all semialgebraic sets are typical examples of \({\mathfrak X}\)-families.

Examples of \({\mathfrak X} \)-families other than the families of subanalytic sets and of semi-algebraic sets are the following.

(a) The family of all locally semialgebraic sets.

(b) An 0-minimal Tarski system generated by the relations \(x+y=z\), \(x\cdot y=z\) and \(\exp (x)=y\). Note that the closure of the graph of the function \(y=\exp (-1/x^2)\) is an element of this system and \(C^\infty\) smooth but not \(C^\omega\) smooth.

(c) Let \(X\subset \mathbb{R}^n\) be a closed semialgebraic set, and let \(f_i\), \(i=1, \dots,k\), be Pfaffian functions on an open neighbourhood \(U\) of \(X\), i.e., for each \(i\) \({\partial f_i \over \partial x_j} (x)=P_j(x,f_i(x))\) on \(U\), \(j=1, \dots,n\), for some polynomial functions \(P_j\). Then each element of the smallest family \({\mathfrak X}\) containing graph \(f_i |_X\) is described by \(f_i\) and a finite number of polynomial functions.

For the entire collection see [Zbl 0812.00016].