I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be:
$A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$
where A and C are constant integer matrices, $\vec{b}$ and $\vec{c}$ are constant vectors in $\mathbb{Z}^d$ and $\vec{x}$ is the unknown vector in $\mathbb{Z}^d$.
In order for the mod to make sense, let us throw in some inequalities as well:
$C \vec{x}+\vec{d} > \vec{1}$ where $\vec{1}$ is the all one vector.
Question: Are these equations decidable? If yes, is there a bound known on the size of their solutions? If no, is there a reduction from Matiyasevich's theorem?