具有有理系数的有限多项式方程组的某些实（复）解的界
交换代数
2010-03-30 v9 代数几何
逻辑
摘要
我们讨论两个猜想。(I) 对每个 x_1,...,x_n \in R (C)，存在 y_1,...,y_n \in R (C) 使得 \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) 设 G 为 C 的加法子群。则对每个 x_1,...,x_n \in G，存在 y_1,...,y_n \in G \cap Q 使得 \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)。
引用
@article{arxiv.math/0702558,
title = {Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients},
author = {Apoloniusz Tyszka},
journal= {arXiv preprint arXiv:math/0702558},
year = {2010}
}
备注
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''