Two conjectures on the arithmetic in ℝ and ℂ

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