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Characterization of quadratic $\varepsilon$-CNS polynomials (extended version)

Borka Jadrijević, Kristina Miletić

Abstract

In this paper, we give characterization of quadratic $\varepsilon-$canonical number system ($\varepsilon-$CNS) polynomials for all values $\varepsilon \in\lbrack0,1)$. Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials ($\varepsilon=0$) and quadratic SCNS polynomials ($\varepsilon=1/2$). This result is a consequence of our new characterization results of $\varepsilon -$shift radix systems ($\varepsilon-$SRS) in the two-dimensional case and their relation to quadratic $\varepsilon-$CNS polynomials.

Characterization of quadratic $\varepsilon$-CNS polynomials (extended version)

Abstract

In this paper, we give characterization of quadratic canonical number system (CNS) polynomials for all values . Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials () and quadratic SCNS polynomials (). This result is a consequence of our new characterization results of shift radix systems (SRS) in the two-dimensional case and their relation to quadratic CNS polynomials.
Paper Structure (5 sections, 20 theorems, 72 equations, 2 figures, 2 algorithms)

This paper contains 5 sections, 20 theorems, 72 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main theorem and basic idea of its proof
  3. Complete characterization of set $\mathcal{D}_{2,\varepsilon}^{0}$ in half-plane $R\left( \varepsilon\right)$
  4. Characterization of quadratic $\varepsilon-$CNS polynomials
  5. Proofs of auxiliary results from Section \ref{['Sect3']}

Key Result

Theorem 1

Let$\varepsilon \in\left[ 0,1\right)$ and $P\left( x\right) =x^{d}+p_{d-1}x^{d-1}+...+p_{1}x+p_{0}\in\mathbb{Z}\left[ x\right]$. Then $P$ is $\varepsilon-$CNS polynomialif and only if $\left( \frac{1}{p_{0}},\frac{p_{d-1}}{p_{0}},...,\frac{p_{1}}{p_{0}}\right) \in\mathcal{D}_{d,\varepsilon}^{0}

Figures (2)

  • Figure 1: The sets $\overline{B\left( \varepsilon\right) }$ and $\overline{T\left( \varepsilon\right) }.$
  • Figure 2: Convex hull $\Delta _{14}$.

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Theorem 1: Surer
  • Theorem 2
  • Proposition 1: Akiyama-Scheicher
  • Corollary 1
  • proof
  • Theorem 3
  • Remark 1
  • Remark 2
  • ...and 39 more