Atypical values at infinity of polynomial and rational functions on an algebraic surface in \(\mathbb R^n\)
The authors give a characterization of the atypical values at infinity for a real rational function \(f/g\) defined on a real algebraic surface \(V\) in \(\mathbb{R}^n\). In order to do so, the authors define a 1--dimensional semi--algebraic subset \(\Gamma\) of \(V\), called tangency curve of \((f/g)|_V\) and consider the half--branches of \(\Gamma\) at infinity. Let \(H=0\) be an algebraic equation for \(\Gamma\) (\(H\) depends on \(f,g\) and the real polynomials defining \(V\)). For each half--branch \(C\) of \(\Gamma\), \(f/g\) is monotonous (increasing, decreasing or constant) over \(C\) and \(H\) might or might not change sign along \(C\). With this information and a point \(K\) to make a start (the authors choose \(K\) to be a connected component of \(V\setminus \{g=0\}\) intersected with a big enough sphere), critical clusters belonging to \(\lambda\in\mathbb R\), bands, valleys and crests are defined. Then, for each critical cluster, an integer in \(\{0,\pm 1,\pm 2\}\) is computed. The main theorem of the paper asserts that \(\lambda\in\mathbb R\) is an atypical value at infinity for \((f/g)|_V\) if and only if there exists a critical cluster belonging to \(\lambda\) such that the corresponding integer is non--zero. The second main result provides an upper bound for the number of atypical values for \((f/g)|_V\), in terms of the degrees of \(f,g\) and the polynomials defining \(V\). The paper finishes with three worked out examples. The techniques used in the proofs are standard of algebraic geometry and real algebraic geometry. Definitions and arguments closely follow those in a paper by \textit{M. Coste} and \textit{M. J. de la Puente} [J. Pure Appl. Algebra 162, No. 1, 23--35 (2001; Zbl 1042.14038)], as the authors themselves say. The paper is rather interesting. Some questions arise, though. For instance, in all the examples contained in the paper, \(g=1\). Is the case \(g\) general much more complicated? Another question is the following. In all the examples in the paper, \(V\) is a hypersurface in \(\mathbb{R}^3\). Is the general case much more involved? And, finally, are the bounds sharp?
scientific article; zbMATH DE number 5989481