Degree of rational mappings, and the theorems of Sturm and Tarski (Q942964)

The paper is an extended version of lecture notes (taken by the second author) of a course given by the first author at the University of Toronto. It gives an algorithm for computing the mapping degree of a rational mapping \(\mathbb{R}\mathbb{P}^1\to\mathbb{R}\mathbb{P}^1,\) which serves as a basic tool for proving Sturm's theorem. The latter in turn is used to prove Tarski's theorem on semialgebraicity of the image of a real semialgebraic set under a polynomial mapping, and the theorem on constructibility of the image of a complex constructible set under a polynomial mapping. References to other modern algorithmic treatments of the latter classical results are missing in the paper -- we can recommend e.g. [\textit{S.~Basu, R.~Pollack} and \textit{M.-F.~Roy}, Algorithms in real algebraic geometry. 2nd ed. Berlin: Springer (2006; Zbl 1102.14041)].