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Keywords:

  • System of polynomial equations;
  • system of linear equations;
  • solution with minimal l norm

Abstract

Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. (1) If a system SEn is consistent over ℝ (ℂ), then S has a real (complex) solution which consists of numbers whose absolute values belong to [0, 22n –2]. (2) If a system SWn is consistent over G, then S has a solution (x1, …, xn) ∈ (G ∩ ℚ)n in which |xj| ≤ 2n –1 for each j.