Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients

Details Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients

2011-08-20

arxiv.org/abs/1108.4078v1

Mathematics

Classical Analysis and ODEs

12 pages

Scientific paper

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer $k$. We show that if the system has a nontrivial solution then there exists a nontrivial solution $\x=(x_1,...,x_n)\trans$ such that $\frac{|x_j|}{|x_i|}\le k^{n-1}$ for each $i,j$ satisfying $x_ix_j\ne 0$. This inequality is sharp. We also prove a conjecture of A. Tyszka related to our results.

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