## Abstract

Consider the inequalities (a) |Ax| ≤ b, A ∈ ℝ^{r x n}_{r}, r < n, b positive vector (here |y| denotes the vector of absolute values of components of the vector y) and (b) x^{T} Ax ≤ α, A positive semi-definite ∈ ℝ^{nxn}_{r},r < n, α. > 0. Both inequalities are guaranteed a nonzero integer solution x for every positive right-hand side (b, α respectively). Such solutions will generally have a nonzero orthogonal projection X_{N(A)} on the null space of A. We prove that a nonzero integer solution x exists with ∥ X_{N(A)} ∥ bounded, for (a): ∥X_{N(A)}∥ ≤ √n-r (vol A/b^{1} ⋯ b_{r}) 1/(n-r), for (b): ∥X_{N(A)}∥ ≤ (2^{n} √ vol A/α^{r/2} K_{n}) 1/(n-r), where vol A = √ Σ det^{2} A_{IJ} summing over all r x r submatrices A_{IJ}, and K_{n} is the volume of the Euclidean unit ball in ℝ^{n}.

Original language | English (US) |
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Pages (from-to) | 279-282 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 169 |

Issue number | 1-3 |

DOIs | |

State | Published - May 15 1997 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

## Keywords

- Convex sets
- Determinants
- Geometry of numbers
- Lattice points
- Singular values

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