Bifurcation values and stability of algebras of bounded polynomials (Q468759)

In the paper under review, a relation between bifurcation values at infinity of a polynomial \(f:\mathbb{R}^2\to \mathbb{R}\) and the algebras of polynomials bounded on \(\{x\in \mathbb{R}^2\mid f(x)\leq c\}\) is presented.NEWLINENEWLINELet us explain the notion of bifurcation values at infinity:NEWLINENEWLINELet \(h:\mathbb{K}^n\to \mathbb{K}\) be a \(C^\infty\)-function where \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\). The function \(h\) is a \textit{trivial \(C^\infty\)-fibration at infinity} over an open set \(U\subset\mathbb{K}\) if for any \(c\in U\) there exists a compact set \(K\subset\mathbb{K}^n\) and a \(C^\infty\)-diffeomorphism \(\phi:h^{-1}(U)\setminus K\to U\times (h^{-1}(c)\setminus K)\) such that \(\pi\circ \phi=h|_{h^{-1}(U)\setminus K}\) where \(\pi\) is the projection on the first coordinate. A number \(c\in \mathbb{K}\) is called \textit{a bifurcation value at infinity} of \(h\) if there is no open neighbourhood \(U\) of \(c\) such that \(h\) is a trivial \(C^\infty\)-fibration at infinity over \(U\). The set \(B_{\mathbb{K}}^\infty(h)\) denotes the set of bifurcation values at infinity of \(h\).NEWLINENEWLINEThe main result is the following:NEWLINENEWLINETheorem. Let \(f\in \mathbb{R}[X,Y]\) and \(0<c<\tilde{c}\in\mathbb{R}\) such that \(B_{\mathbb{C}}^\infty(f)\cap [c,\tilde{c}]=\emptyset\). Then \(g\in \mathbb{R}[X,Y]\) is bounded on \(\{f\leq \tilde{c}\}\) if \(g\) is bounded on \(\{f\leq c\}\).