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A generalized number system (GNS for short) over $\\mathcal{O}$ is a pair $(p,\\mathcal{D} )$ where $p\\in\\mathcal{O}[x]$ is monic with constant term $p(0)$ not a zero divisor of $\\mathcal{O}$, and where $\\mathcal{D}$ is a complete residue system modulo $p(0)$ in $\\mathcal{O}$ containing $0$. 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