On analytic equivalence of functions at infinity
We say that two functions \(f,g:\mathbb{R}^n\to \mathbb{R}\) are analytically equivalent at infinity if there exist \quad 1. an analytic diffeomorphism \(\phi \) of neighbourhoods of infinity in \(\mathbb{R}^{n}\) satisfying the condition \quad \quad\; \(\| \phi (x)\| \to \infty\) if and only if \(\| x\| \to \infty\), \quad 2. an analytic diffeomorphism \(\psi :\mathbb R\to\mathbb R\), such that \(f\circ \phi =\psi \circ g\) in a neighbourhood of \quad \quad\; infinity in \(\mathbb{R}^{n}\). The main result of the paper is the following: If \(f\in \mathbb R[x_1,\dots ,x_n ]\) satisfies the Łojasiewicz inequality \(\| \nabla f(x)\| \geq C\| x\| ^{k-1}\) in a neighbourhood of infinity in \(\mathbb{R}^{n}\) for some constants \(C>0\) and \(k\in \mathbb{Z}\), \(k\geq 0\), then there exists \(\varepsilon >0\) such that for each polynomial \(P\in \mathbb R[x_1,\dots ,x_n]\), \(\deg p\leq k\), for which the absolute values of the coefficients of the monomials of \(P\) of degree \(k\) do not exceed \(\varepsilon\), the functions \(f\) and \(f+P\) are analytically equivalent at infinity. The result is also generalised to the Nash functions, to \(C^{r+1}\) functions, and to the complex case.
scientific article; zbMATH DE number 5934206