Let $a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}=0$ be a homogenous linear Diophantine equation on $n$ variables. Are there any conditions for this equation to have non trivial solutions? And if so, how can we calculate them?
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1See e.g. here – Bill Dubuque Jun 13 '23 at 13:45