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Quantum algorithms for the Sylvester denumerant and the numerical semigroup membership problem

J. Ossorio-Castillo, José M. Tornero

Abstract

Two quantum algorithms are presented, which tackle well--known problems in the context of numerical semigroups: the numerical semigroup membership problem (NSMP) and the Sylvester denumerant problem (SDP).

Quantum algorithms for the Sylvester denumerant and the numerical semigroup membership problem

Abstract

Two quantum algorithms are presented, which tackle well--known problems in the context of numerical semigroups: the numerical semigroup membership problem (NSMP) and the Sylvester denumerant problem (SDP).
Paper Structure (7 sections, 6 theorems, 24 equations, 6 figures)

This paper contains 7 sections, 6 theorems, 24 equations, 6 figures.

Table of Contents

  1. Numerical Semigroups
  2. The Library
  3. Sylvester Denumerant and Numerical Semigroup Membership
  4. $\mathbb{SETUP}$
  5. $\mathbb{STEP}$ 1
  6. $\mathbb{STEP}$ 2
  7. Conclusions

Key Result

Lemma 1.3

Let $A = \{a_1,...,a_n\}$ be a nonempty subset of $\mathbb{Z}_{\geq 0}$. Then, is a numerical semigroup if and only if $\gcd(a_1, ... , a_n) = 1$.

Figures (6)

  • Figure 1: Representation of the values of $\lambda_i$ with respect to the $b$ qubits
  • Figure 2: output for $d(10000; 376,381,393,399)$
  • Figure 3: Sylvester quasi-polynomial
  • Figure 4: Sylvester quasi-polynomial (detail)
  • Figure 5: Sylvester quasi-polynomial (detail)
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10