A polynomial time algorithm for Sylvester waves when entries are bounded

The Sylvester's denumerant \ (d (t; a) \) is a quantity that counts the number of nonnegative integer solutions to the equation \ (₈=₁^N aᵢ xᵢ = t \), where \ (a = (a₁, , aN) \) is a sequence of distinct positive integers with \ ( (a) = 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 \ (aᵢ 500 \) compared to Sills-Zeilberger's Maple package PARTITIONS.