{"paper":{"title":"Upper bounds on the magnitude of solutions of certain linear systems with integer coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Gaspar Porta, Pedro J. Freitas, Shmuel Friedland","submitted_at":"2011-08-20T02:00:41Z","abstract_excerpt":"In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals.\n  Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer\n  $k$. 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.\n  We also prove a conjecture of A. Tyszka related to our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4078","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}