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