We are interested in computing alternate sums of Euler characteristics of some particular semialgebraic sets, intersections of an algebraic one, smooth or with finitely many singularities, with sets given by just one polynomial inequality. We state theorems relating these alternate sums of characteristics to some topological degrees at infinity of polynomial mappings.
"Degree formulas for the Euler characteristic of semialgebraic sets." Hokkaido Math. J. 48 (3) 461 - 473, October 2019. https://doi.org/10.14492/hokmj/1573722013