Degree formulas for the Euler characteristic of semialgebraic sets (Q2280565)

Let \(F = (F_1, \ldots, F_k) \colon \mathbb{R}^n \to \mathbb{R}^k\) and \(G = (G_1, \ldots, G_l) \colon \mathbb{R}^n \to \mathbb{R}^l\) be polynomial maps with \(k, l \ge 1.\) Consider the semialgebraic sets: \[W_G(\epsilon) := \{x \in \mathbb{R}^n : F_i(x) = 0, i = 1, \ldots, k\ \text{ and } \ (-1)^{\epsilon_j} G_j(x) \ge 0, j = 1, \ldots, l\},\] where \(\epsilon := (\epsilon_1, \ldots, \epsilon_l) \in \{0, 1\}^l.\) Define the polynomial map \(L \colon \mathbb{R}^n \times \mathbb{R}^k \times \mathbb{R}^l \to \mathbb{R}^n \times \mathbb{R}^k \times \mathbb{R}^l\) by \[L(x, \lambda, \mu) := \left(x + \sum_{i = 1}^k \lambda_i \nabla F_i(x) + \sum_{j = 1}^l \mu_j \nabla G_j(x), F(x), \mu_1 G_1(x), \ldots, \mu_l G_l (x) \right).\] Using the Morse theory for manifolds with corners, the author shows, under a transversality condition, that \[\sum_{\epsilon \in \{0, 1\}^l} (-1)^{|\epsilon|} \chi(W_G(\epsilon)) = (-1)^k \deg_\infty L,\] where \(|\epsilon| := \sum_{j = 1}^l \epsilon_j,\) \(\chi(\cdot)\) stands for the Euler characteristic and \(\deg_\infty L\) stands for the topological degree at infinity of \(L.\)