Special kind of a linear Linear Diophantine equation

Could any one help me to point out some literature/ papers which solves a homogenous linear Diophantine equation (one equation) of the form

$a_1 \times x_1+a_2 \times x_2 + a_3 \times x_3+....+a_n \times x_n=0$, where $a_1,a_2,...,a_n$ are positive or negative integer constants and $x_1,x_2,...,x_n$ are positive integer variables.

Such an equation may have no solution or infinite number of solutions however there must exist be some minimal solutions or a set of base solutions which can be used to derive ALL other solutions of the equation.

1) Could you point out some work/literature which tries to find the set of base solutions of the above equation.

• Reduce to the case where the $x_i$ are coprime and go from there. – Adam Hughes Jun 30 '14 at 5:27