A polynomial time algorithm for Sylvester waves when entries are bounded

Guoce Xin; Chen Zhang

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arXiv · 2406.18975

A polynomial time algorithm for Sylvester waves when entries are bounded

Guoce Xin·Chen Zhang

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Abstract

The Sylvester's denumerant $ d(t; \boldsymbol{a}) $ is a quantity that counts the number of nonnegative integer solutions to the equation $ \sum_{i=1}^{N} a_i x_i = t $, where $ \boldsymbol{a} = (a_1, \dots, a_N) $ is a sequence of distinct positive integers with $ \gcd(\boldsymbol{a}) = 1 $. We present a polynomial time algorithm in $N$ for computing $ d(t; \boldsymbol{a}) $ when $ \boldsymbol{a} $ is bounded and $ t $ is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in \texttt{Maple} under the name \texttt{Cyc-Denum} and demonstrates superior performance when $ a_i \leq 500 $ compared to Sills-Zeilberger's \texttt{Maple} package \texttt{PARTITIONS}.

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BibTeX RIS

Guoce Xin, Chen Zhang. 2024-06-27. A polynomial time algorithm for Sylvester waves when entries are bounded. https://arxiv.org/abs/2406.18975

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