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.