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

Abstract

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)

LaTeX2e, 28 pages, a shortened and revised version will appear in Mathematical Logic Quarterly 56 (2010), no.2, under the title ``Two conjectures on the arithmetic in R and C''

References in corpus (5)

• Characterizing integers among rational numbers with a universal-existential formula
• Some conjectures on addition and multiplication of complex (real) numbers
• Arithmetic neighbourhoods of numbers
• A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity
• A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity

Cited by in corpus (4)

• Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients
• A subset of Z^n whose non-computability leads to the existence of a Diophantine equation whose solvability is logically undecidable
• Small systems of Diophantine equations which have only very large integer solutions
• Two conjectures on integer arithmetic and their applications to Diophantine equations