Given z∈C{sup}n and A∈Z{sup}(m×n), we provide an explicit expression and an algorithm for evaluating the counting function h(y; z) :=∑{z{sup}x|x∈Z{sup}n; Ax=y, x≥0}. The algorithm only involves simple (but possibly numerous) calculations. In addition, we exhibit finitely many fixed convex cones of R{sup}n explicitly and exclusively defined by A, such that for any y∈Z{sup}m, h(y; z) is obtained by a simple formula that evaluates ∑z{sup}x over the integral points of those cones only. At last, we also provide an alternative (and different) formula from a decomposition of the generating function into simpler rational fractions, easy to invert.
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机译：给定z∈c{sup} n和a∈z{sup}（m×n），我们提供了一种显式表达式和用于评估计数函数h（y; z）的算法：=σ{z {sup} x | x∈z{sup} n; x = y，x≥0}。该算法仅涉及简单（但可能众多）的计算。另外，我们明确地展现了r {sup} n的许多固定凸锥体，例如由a，例如任何Y∈z{sup} m，h（y; z）通过评估σ的简单公式获得z {sup} x仅在那些锥体的整数点上。最后，我们还提供了一种替代（和不同）公式从发电功能的分解成更简单的合理分数，易于反转。
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