Counting real roots of polynomial endomorphisms. (Q1874986)

The first three sections of the paper under review have a survey's character. The first one is devoted to recall some sufficient conditions for a polynomial vector field \(f:\mathbb{R}^n\to\mathbb{R}^n\) to be proper. The interested reader is also refered to the work by \textit{C. Ueno} and the reviewer [in: Real algebraic geometry, Lect. Notes Math. 1524, 240--256 (1992; Zbl 0793.14038)]. The second one collects today's classical results by \textit{E. Becker} and \textit{T. Wörmann} [in: Recent advances in real algebraic geometry and quadratic forms. Contemp. Math. 155, 271--291 (1994; Zbl 0835.11016)] and by \textit{P. Pedersen}, \textit{M.-F. Roy} and \textit{A. Szpirglas} [in: Computational algebraic geometry. Prog. Math. 109, 203--224 (1993; Zbl 0806.14042)] which lead to effective formulae to compute the cardinal of the fibers and the topological degree of some proper polynomial maps. The particular case of nondegenerated homogeneous polynomial vector fields with fixed multi-degree is explained in Section 3, by following, essentially, two papers of \textit{G. N. Khimshiashvili} [Soobshch. Akad. Nauk Gruz. SSR 85, 309--312 (1977; Zbl 0346.55008); Georgian Math. J. 8, No. 1, 97--109 (2001; Zbl 0987.32013)]. A very nice and clear article on this topic is due to \textit{A. Lecki} and \textit{Z. Szafraniec} [in: Topology in nonlinear analysis. Banach Cent. Publ. 35, 73--83 (1996; Zbl 0872.55004)]. The most original part of the paper is Section 4, which concerns the bidimensional case, with emphasis on the effectiveness of the proposed methods. Sometimes along the paper the conditions imposed to the vector field \(f\) are unclear for the reviewer. First, notice that in the real case properness does not imply finiteness of the fibers; for example, many fibers of the proper polynomial map \[ f: \mathbb{R}^2\to\mathbb{R}^2: (x,y)\to (x^2+y^2,x^2+y^2) \] are circumferences. So, it has no meaning to compute ``the cardinal of the fibers'' in such a case. On the other hand, the author seems to correct this in the last line of page 5329 by imposing \(f\) to be smooth. The reviewer has not found the meaning of smooth in this context along the paper; perhaps it could mean that the Jacobian determinant of \(f\) does not vanish at any point of \(\mathbb{R}^n\). But this, together with the properness of \(f\), implies that \(f\) is a diffeomorphism, by Hadamard theorem, and so the fibers are singletons.