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A polynomial time algorithm for Sylvester waves when entries are bounded

Guoce Xin, Chen Zhang

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}.

A polynomial time algorithm for Sylvester waves when entries are bounded

Abstract

The Sylvester's denumerant \( d(t; \boldsymbol{a}) \) is a quantity that counts the number of nonnegative integer solutions to the equation , 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 for computing \( d(t; \boldsymbol{a}) \) when is bounded and 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 compared to Sills-Zeilberger's \texttt{Maple} package \texttt{PARTITIONS}.

Paper Structure

This paper contains 8 sections, 10 theorems, 42 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Computation of $W_f(t; \boldsymbol{a})$
  3. Two Tools
  4. Computation of $W_f(t; \boldsymbol{a})$
  5. Algorithm and complexity analysis
  6. Experiments
  7. Floating computation
  8. Concluding remark

Key Result

Lemma \oldthetheorem

Let $f$ be a positive integer, and let $F(x)$ be a rational function in $x$. Suppose $F(\zeta)$ exists for all $\zeta$ satisfying $\zeta^f=1$. Then

Figures (2)

  • Figure 1: Running time (in seconds) for computing $d(t; 1, 2, \dots, k)$.
  • Figure 2: The ratio of running time for computing $d(t; 1, 2, \dots, k)$ between different algorithms.

Theorems & Definitions (21)

  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem: 2023-Xin-GTodd
  • Lemma \oldthetheorem: 2023-Xin-GTodd
  • Lemma \oldthetheorem: 2023-Xin-GTodd
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • ...and 11 more