Multi-degree bounds on the Betti numbers of real varieties and semi-algebraic sets and applications

摘要: We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence degrees polynomials defining them than results known before. Our method also unifies several different types under single framework, such as depending total degrees, multi-degrees, well in case quadratic partially polynomials. The we present offer significant improvement over what was previously known. Finally, extend result Barone Basu bounding number connected components defined by two differing to sum all numbers, thus making progress an open problem posed paper.