Fq -Zeros of Sparse Trivariate Polynomials and Toric 3-Fold Codes

For a given lattice polytope P ⊂ R3, consider the space L P of trivariate polynomials over a finite field Fq, whose Newton polytopes are contained in P. We give an upper bound for the maximum number of Fq-zeros of polynomials in L P in terms of the Minkowski length of P and q, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of L P at the points of the algebraic torus (F*q )3. Our approach is based on understanding factorizations of polynomials in L P with the largest possible number of nonunit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in P with the largest possible number of nontrivial summands.