Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients
Abstract
We discuss two conjectures. (I) For each x1,...,xn ∈ R (C) there exist y1,...,yn ∈ R (C) such that ∀ i ∈ 1,...,n |yi| ≤ 22n-2 ∀ i ∈ 1,...,n (xi=1 ⇒ yi=1) ∀ i,j,k ∈ 1,...,n (xi+xj=xk ⇒ yi+yj=yk) ∀ i,j,k ∈ 1,...,n (xi · xj=xk ⇒ yi · yj=yk) (II) Let G be an additive subgroup of C. Then for each x1,...,xn ∈ G there exist y1,...,yn ∈ G Q such that ∀ i ∈ 1,...,n |yi| ≤ 2n-1 ∀ i ∈ 1,...,n (xi=1 ⇒ yi=1) ∀ i,j,k ∈ 1,...,n (xi+xj=xk ⇒ yi+yj=yk)
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