The index of isolated point of the flow on the real 2-dimensional semi-algebraic sets (Q2767619)

Let \(P(x_1,\ldots,x_{n})\) be a polynomial. A set in \(\mathbb R^{n}\) is called semi-algebraic if it has a representation as a finite union of intersections of sets of the form \(\{(x_1,\ldots,x_{n})|P(x_1,\ldots,x_{n})=0\}\) or \(\{(x_1,\ldots,x_{n})|P(x_1,\ldots,x_{n})<0\}\). The authors prove that if on a compact semi-algebraic set \(M\) a flow with isolated singular points \(x_{i}\) is given, then \(\chi(M)=\sum \text{ind} (x_{i})\), where \(\chi(M)\) is the Euler characteristic of \(M\); \(\text{ind} (x_{i})\) is the index of the singular point \(x_{i}\).