A polynomial time algorithm for Sylvester waves when entries are bounded

Abstract

Sylvester's denumerant $d(t;a)$ is a quantity that counts the number of nonnegative integer solutions to the equation ${\sum }_{i=1}^N{a}_i{x}_i=t$, where $a=(a_1,a_2,\dots ,a_N)$ is a sequence of positive integers with $\gcd \left(a\right)=1$. We present a polynomial time algorithm in N for computing $d(t;a)$ when 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 Maple under the name Cyc-Denum and demonstrates superior performance when $a_i\le 500$ compared to Sills-Zeilberger's Maple package PARTITIONS.