Characterization of quadratic \(\varepsilon\)-CNS polynomials (Q6556232)

Let \(P(x)= x^d+p_{d-1}x^{d-1}+\cdots+p_1x + p_0 \in \mathbb Z[x]\), \(|p_0|\geq 2\), \(\varepsilon\in [0,1)\). The number system \((P, \mathcal N_\varepsilon)\) is called an \(\varepsilon\)-canonical number system (CNS for short) if\N\begin{align*}\N\mathcal N_\varepsilon=[-\varepsilon|p_0|,(1-\varepsilon)|p_0|)\cap\mathbb Z.\N\end{align*}\N\(P(x)\) is called the basis of the \(\varepsilon\)-CNS or \(\varepsilon\)-CNS polynomial. \(P(x)\) is called the \(\varepsilon\)-set of digits.\N\NThe paper under review mainly provides a characterization of quadratic \(\varepsilon\)-canonical number system (\(\varepsilon\)-CNS) polynomials for all values \(\varepsilon\in [0,1)\). More precisely, the authors mainly prove the following result:\N\NLet \(P(x)= x^2+p_1x + p_0 \in \mathbb Z[x]\), \(|p_0|\geq 2\), and \(\varepsilon\in [0,1)\). Let \(k=\lfloor \varepsilon|p_0|\rfloor\). Then the corresponding \(\varepsilon\)-set of digits is \(\mathcal N_\varepsilon=\{-k,\ldots, |p_0|-1-k\}\).\N\Ni) Let \(\varepsilon\in [0,\frac{1}{2})\) or let \(\varepsilon=\frac{1}{2}\) when \(|p_0|\) is odd. Then \(P(x)\) is an \(\varepsilon\)-CNS polynomial if and only if\N\begin{align*}\N-k-1\leq p_1 \leq |p_0|-k, \quad p_0\geq 2, \quad\text{for}\quad\varepsilon\in \left[0,\frac{1}{2}\right),\N\end{align*}\Nor\N\begin{align*}\Nk+2-|p_0|\leq p_1 \leq k-1, \quad p_0\leq -3, \quad\text{when}\quad\varepsilon\in \left[\frac{1}{|p_0|},\frac{1}{2}\right).\N\end{align*}\N\Nii) Let \(\varepsilon\in (\frac{1}{2},1)\) or let \(\varepsilon=\frac{1}{2}\) when \(|p_0|\) is even. Then \(P(x)\) is an \(\varepsilon\)-CNS polynomial if and only if\N\begin{align*}\N-|p_0|+k\leq p_1 \leq k+1, \quad p_0\geq 2, \quad\text{for}\quad\varepsilon\in \left[\frac{1}{2},1\right),\N\end{align*}\Nor\N\begin{align*}\N-k+1\leq p_1 \leq -k-2+|p_0|, \quad p_0\leq -3, \quad\text{when}\quad\varepsilon\in \left[\frac{1}{2},\frac{|p_0|-1}{|p_0|}\right).\N\end{align*}