A polynomial time algorithm for Sylvester waves when entries are bounded
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929401815367680 |
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| author | Xin, Guoce Zhang, Chen |
| author_facet | Xin, Guoce Zhang, Chen |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18975 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A polynomial time algorithm for Sylvester waves when entries are bounded Xin, Guoce Zhang, Chen Combinatorics Primary 05--08, Secondary 05--04, 05A17 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}. |
| title | A polynomial time algorithm for Sylvester waves when entries are bounded |
| topic | Combinatorics Primary 05--08, Secondary 05--04, 05A17 |
| url | https://arxiv.org/abs/2406.18975 |