On sums of Betti numbers of affine varieties

TL;DR

This paper establishes an upper bound on the sum of Betti numbers for affine varieties defined by polynomials of bounded degree, answering a question posed by Katz in 2001.

Contribution

It provides a new explicit bound on Betti numbers of affine varieties, advancing understanding in algebraic geometry and topology.

Findings

Bound on Betti numbers: 2(N+1)^{2N+1}(d+1)^N

Answers Katz's 2001 question

Applicable to varieties defined by degree d polynomials

Abstract

We show that if V is a subvariety of the affine N-space defined by polynomials of degree at most d, then the sum of its $\ell$-adic Betti numbers does not exceed $2(N + 1)^{2N +1}(d+ 1)^N$. This answers a question of Katz (FFA 2001).