Hausdorff measure of zeros of nontrivial multivariable polynomials

This paper establishes a sharp universal bound on the $(d-1)$-dimensional Hausdorff measure of the zero level set of any nontrivial multivariable polynomial $p:R^d\to R$ within d-dimensional cubes of size r. The bound depends only on the cube size r, the dimension d, and the degrees of the polynomial p.