The paper deals with finiteness problems concerning semialgebraic and Nash sets, nad Nash functions. A subset \(X\subset \mathbb{R}^n\) is semialgebraic when it has a description by finite boolean combination of polynomial equations and inequalities. An affine Nash manifold is a pure dimensional semialgebraic subset \(M\subset \mathbb{R}^n\) that is a smooth submanifold of an open subset of \(\mathbb{R}^n\). A Nash function on an open semialgebraic set \(U\subset M\) is a semialgebraic smooth function on \(U\). A Nash subset of \(U\) is the zero set of a Nash function on \(U\). Nash functions are also considered in more general case. A Nash function on a semialgebraic set \(X\subset M\) is a cross-section over \(X\) of the sheaf of germs of Nash functions on Nash manifold \(M\).
The results of the paper are within the range of research of comparison the Euclidean and semialgebraic topology initiated by G. W. Brumfiel [Partially ordered rings and semi-algebraic geometry. Cambridge etc.: Cambridge University Press (1979; Zbl 0415.13015)]. Finiteness problems arise in connection with the fact that the semialgebraic topology is not a true topology. For instance, any Nash function \(f:X\to\mathbb{R}\) on a semialgebraic set \(X\subset M\) can be extended to an analytic function defined on an open set \(U\subset M\) in the Euclidean topology. The set \(U\) as infinite union of open semialgebraic sets is not necessary semialgebraic neighbourhood of \(X\). Therefore, a problem arises, if we can find such an extension, which is defined on an open semialgebraic set. The authors obtain an affirmative answer to this question in Theorem 1.3.
The remaining results of the article are related to possibility of finite description for properties of local nature in Euclidean topology. Among other results, it is shown that: a Nash set \(X\) that has only normal crossings in \(M\) can be covered by finitely many open semialgebraic sets \(U\) equipped with Nash diffeomorphisms \((u_1,\dots,u_m):U\to \mathbb{R}^m\) such that \(U\cap X=\{u_1\cdots u_r=0\}\) for some \(r\) (Theorem 1.6, Theorem 1.7 has a similar nature). Every affine Nash manifold with corners \(N\) is a closed subset of an affine Nash manifold \(M\) where the Nash closure of the boundary \(\partial N\) of \(N\) has only normal crossings and \(N\) can be covered with finitely many open semialgebraic sets \(U\) such that each intersection \(N\cap U=\{u_1\geq 0,\dots u_r\geq 0\}\) for a Nash diffeomorphism \((u_1,\dots,u_m):U\to \mathbb{R}^m\) (Theorem 1.11).