Digit systems in polynomial rings over finite fields
Let \(\mathbb F\) be a finite field and \(p(x,y) = \sum_{j=0}^m b_j(x)y^j \in {\mathbb F}[x,y]\). Furthermore, let \({\mathcal N} = \{p \in {\mathbb F}[x] : \deg p(x) < \deg b_0(x)\}\). Then \(p(x,y)\) is called a DS-polynomial if every element of the quotient ring \(R = {\mathbb F}[x,y]/p(x,y){\mathbb F}[x,y]\) has a finite representation \[ q = d_0(x) + d_1(x)y + \dots + d_k(x)y^k \] with \(d_i(x) \in \mathcal N\), \(0 \leq i \leq k\). In this paper all DS-polynomials are characterized. It is proved that if \(\deg b_n = 0\) and \(\deg b_0 > 0\) then \(p(x,y)\) is a DS-polynomial if and only if \(\deg b_i<\deg b_0\) for all \(1\leq i \leq m\). This result is interesting because the concept of DS-polynomials is a generalization of CNS polynomials, which was introduced in [\textit{A. Pethő}, Computational number theory, Proc. Colloq., Debrecen 1989, 31-43 (1991; Zbl 0733.94014)]. In contrast to the very simple characterization of the DS-polynomials, there is no idea for a similar result even for cubic CNS polynomials.
scientific article; zbMATH DE number 1965047