\mathbbF_q-zeros of sparse trivariate polynomials and toric 3-fold codes

2021/05/21 by Kyle Meyer, Meyer, Kyle, Ivan Soprunov +3
Mathematics · #11T06 #14G50 #52B10 #52B20 (Primary) 11T71 #52B55 (Secondary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.CO #math.NT #msc:11T06 #msc:11T71 #msc:14G50 #msc:52B10 #msc:52B20 #msc:52B55

paper · pdf · doi:10.48550/arxiv.2105.10071

31 pages; new section on 3-fold toric codes from width one polytopes; new examples of 3-fold toric codes whose parameters exceeds the Gilbert-Varshamov bound; 2 figures

arxiv created 2022/01/28 · arxiv updated 2022/02/01

Abstract

For a given lattice polytope P in ℝ³, consider the space L_P of trivariate polynomials over a finite field \mathbbF_q, whose Newton polytopes are contained in P. We give an upper bound for the maximum number of \mathbbF_q-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 (\mathbbF_q^*)³. Our approach is based on understanding factorizations of polynomials in L_P with the largest possible number of non-unit 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 non-trivial summands.

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