Two conjectures on the arithmetic in ℝ and ℂ†

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 S ⊆ En 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 S ⊆ Wn is consistent over G, then S has a solution (x1, …, xn) ∈ (G ∩ ℚ)n in which |xj| ≤ 2n –1 for each j.