Papers › 𝔽_q-zeros of sparse trivariate polynomials and toric 3-fold codes
𝔽_q-zeros of sparse trivariate polynomials and toric 3-fold codes
Kyle Meyer, Ivan Soprunov, Jenya Soprunova
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
For a given lattice polytope P in ℝ³, consider the space ℒ_P of trivariate polynomials over a finite field 𝔽_q, whose Newton polytopes are contained in P. We give an upper bound for the maximum number of 𝔽_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 (𝔽_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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections