Refined Estimates on the Betti Numbers of Real Algebraic Varieties and Semi-Algebraic Sets
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. In the case of bounded total degree, we obtain a bound on the sum of the Betti numbers with an improved leading coefficient, extending a similar result that bounded only the number of connected components. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known, particularly in the case when the description of the set is given by both polynomials with bounded total degree and others that are partially quadratic. We conclude with some useful applications to discrete geometry that follow from our main results. In the applications presented, we note that our method allows us to explicitly see the dependence on the different degrees involved for the first time.
Basu, Purdue University.
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