Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients

Details Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients

2007-02-19

arxiv.org/abs/math/0702558v86

Mathematical Logic Quarterly 56 (2010), no.2, pp.175-184.

Mathematics

Commutative Algebra

LaTeX2e, 28 pages, a shortened and revised version will appear in Mathematical Logic Quarterly 56 (2010), no.2, under the titl

Scientific paper

10.1002/malq.200910004

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k) \forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k) (II) Let G be an additive subgroup of C. Then for each x_1,...,x_n \in G there exist y_1,...,y_n \in G \cap Q such that \forall i \in {1,...,n} |y_i| \leq 2^{n-1} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)

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