A Sharp Bound for Positive Solutions of Homogeneous Linear Diophantine Equations

Bounds on Positive Integral Solutions of Linear Diophantine Equations

Assuming the existence of a solution, we find bounds for small solutions x of the finite matrix equation Ax = B, where each entry of A, B is an integer, and x is a nontrivial column vector with nonnegative integer entries. 0. Introduction. In [1], [5] and [6] there arise in a topological setting, systems of linear equations with integer coefficients. The problem is to find a bound K depending o...

Optical solutions for linear Diophantine equations

Determining whether a Diophantine equation has a solution or not is the most important challenge in solving this type of problems. In this paper a special computational device which uses light rays is proposed to answer this question, namely check the existence of nonnegative solutions for linear Diophantine equations. The way of representation for this device is similar to an directed graph, h...

Small solutions of linear Diophantine equations

Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <...

Sparse Solutions of Linear Diophantine Equations

We present structural results on solutions to the Diophantine system Ay = b, y ∈ Z ≥0 with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel’s Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

Linear Homogeneous Diophantine Equations and Magic Labelings of Graphs