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arXiv · 2608.14985

A polynomial time algorithm for almost bounded denumerant

Guoce Xin·Chen Zhang·Zihao Zhang
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Abstract

Sylvester's denumerant $d(t; \boldsymbol{A})$ counts the number of nonnegative integer solutions to $\sum_{i=1}^{N} a_i x_i = t$, where $\boldsymbol{A} = (a_1, \dots, a_N)$ is a sequence of positive integers with $\gcd(\boldsymbol{A}) = 1$. In 2025, Xin and Zhang gave a polynomial time algorithm in $N$ for computing $d(t; \boldsymbol{A})$ when the entries of $\boldsymbol{A}$ are bounded by a constant. In this paper, we extend this algorithm by incorporating Barvinok's algorithm, enabling it to handle the case where a fixed number of entries of $\boldsymbol{A}$ are allowed to be unbounded.

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Guoce Xin, Chen Zhang, Zihao Zhang. 2026-08-15. A polynomial time algorithm for almost bounded denumerant. https://arxiv.org/abs/2608.14985

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