2018/10/23 by Evertse, Jan-Hendrik, Győry, Kálmán, Pethő, Attila +1
#11A63 #11R04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1810.09710
Let O be an order, that is a commutative ring with 1 whose additive structure is a free ℤ-module of finite rank. A generalized number system (GNS for short) over O is a pair (p,D ) where p\inO[x] is monic with constant term p(0) not a zero divisor of O, and where D is a complete residue system modulo p(0) in O containing 0. We say that (p,D) is a GNS over O with the finiteness property if all elements of O[x]/(p) have a representative in D[x] (the polynomials with coefficients in D). Our purpose is to extend several of the results from a previous paper of Pethő and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order O and GNS (p,D) over O, the pair (p,D) admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let F be a fundamental domain for O ⊗ℤ ℝ/O and p∈ O[X] a monic polynomial. For α\inO, define pα(x):=p(x+α) and DF ,p(α):= p(α)F\capO. Under mild conditions we show that the pairs (pα,DF,p(α) ) are GNS over O with finiteness property provided α\inO in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that (p-m,DF ,p(-m) ) does not have the finiteness property for each large enough positive rational integer m.