Generalization of the Chinese remainder theorem
Let \(B\) denote an \(m \times (m+1)\) \textit{basis} matrix with integer entries, and let \(r\) denote a column vector with \(m\) integer components. This paper presents sufficient conditions for vector solutions \(x\) with integer components of the linear system \(Bx = r\) in the cases that \(B\) is \textit{marginal} as well as \textit{saturated.} In the marginal case an integer solution exists, if all nonzero entries of \(B\) are pairwise coprime. This immediately generalizes the situation in the classical Chinese remainder theorem given in its matrix form. The proof cleverly involves the notion and basic features of the permanent of matrices. It should be mentioned that the final part of the statement of Theorem 1, in order to describe what is really meant, should be replaced by ``\dots and its \(i_0\)th row is permuted with the leading row of the new matrix''.
scientific article; zbMATH DE number 5345537