Roots and coefficients of multivariate polynomials over finite fields

•We study the number of roots of multivariate polynomials over finite fields.•We generalize results by Kopparty and Wang who studied univariate polynomials.•It is shown that certain patterns of zero coefficients imply few roots.•For products of linear factors certain patterns imply a specific structure on the roots.

Kopparty and Wang studied in [3] the relation between the roots of a univariate polynomial over FqFq and the zero–nonzero pattern of its coefficients. We generalize their results to polynomials in more variables.