A polynomial time algorithm for Sylvester waves when entries are bounded

TL;DR

This paper introduces a polynomial time algorithm for computing Sylvester's denumerant when the entries are bounded, leveraging cyclotomic polynomials and recent advances in Todd polynomial computation.

Contribution

It presents a novel polynomial time algorithm for Sylvester's denumerant with bounded entries, improving computational efficiency for large parameters.

Findings

Algorithm is efficient for entries up to 500.

Implementation in Maple outperforms existing packages.

Demonstrates practical applicability for bounded integer solutions.

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}.