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 \( ∑ᵢ₌₁N a_i x_i = t \), where \( a = (a_1, , a_N) \) 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 \( a_i ≤ 500 \) compared to Sills-Zeilberger's Maple package PARTITIONS.