Hausdorff measure of zeros of polynomials

Abstract

The paper provides an elementary proof establishing a sharp universal bound on the $(d-1)$-Hausdorff measure of the zeros of any nontrivial multivariable polynomial $p:{\mathbb{R}}^d\to \mathbb{R}$ within a $d$-dimensional cube of size $r$. This bound depends solely on the parameter $r$, the dimension $d$, and the degrees of $p$.