An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields

Recently the authors [12] showed that the algebraic integers of the form $${-m+\zeta_k}$$ are bases of a canonical number system of $${\mathbb {Z}[{\zeta_k]}}$$ provided $${m \geqq \phi(k)+1}$$ , where $${\zeta_k}$$ denotes a k-th primitive root of unity and $${\phi}$$ is Euler’s totient function. In this paper we are interested in the questions whether two bases $${-m+\zeta_k}$$ and $${-n+\zeta_k}$$ are multiplicatively independent. We show the multiplicative independence in case that 0 1.

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $\Phi_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. We also obtain a formula for $\Phi_n'(e^{2\pi i/m}) / \Phi_n(e^{2\pi i/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of their multiplicative independence. In the last part we state an appropriate variant of Cobham’s theorem.

Recursive digital filter modelling is one of the tasks, which modelling can be improved by using hypercomplex numbers. Existing models are about data representation in canonical hypercomplex number system only. However, canonical number systems have some restrictions. Applying the non-canonical number systems gives more possibilities for filter simulation and its further optimization by its parametric sensitivity since they have more structure constants in Keli table.

Fractal tiles associated with shift radix systems

In this paper, the so-called compressed Chebyshev polynomials are studied, which are obtained from the original Chebyshev polynomials by suitable scaling transformations. By introducing the concept of cyclotomic pre-polynomials that are in one-to-one correspondence with the well known cyclotomic polynomials, explicit formulas for the irreducible factorization of all four kinds of compressed Chebyshev polynomials are developed. The correspondence between the cyclotomic pre-polynomials and cyclotomic polynomials is based on a polynomial transformation. The same transformation maps all four kinds of the compressed Chebyshev polynomials into polynomials having the property that all their roots are primitive roots of unity, and consequently they are easy to factorize. Generalizations of the applied methods, suitable for the factorization of similar classes of polynomials are also given in the paper.

Cyclotomic polynomials and units in cyclotomic number fields

Interior components of a tile associated to a quadratic canonical number system

The Sum of Digits Function in Number Fields: Distribution in Residue Classes

Signed-digit number system is an unconventional number system which has been found to cause greater efficacy and efficiency in performing some arithmetic operations. Obviously, it seems interesting to investigate signed-digit number system in wider context, including designing magnitude comparator. Although magnitude comparator is an integral part of every standard arithmetic processor, no significant work on magnitude comparison has been reported yet even for the more propitious categories of signed-digit number system, including canonical signed-digit number system. In this paper, an algorithm for magnitude comparison of canonical signed-digit number system is proposed in terms of comparing the magnitudes and signs of some counterpart portions of the inputs. The proposed algorithm can be immediately implemented at hardware level employing some ordinary logic gates only and in every execution stage of the proposed algorithm, all operations may be completed in a fairly small number of logic levels.KeywordsComputer ArithmeticCanonical Signed-Digit NumberMagnitude ComparisonRecurrence RelationPropagation Delay

Generalized radix representations and dynamical systems. IV

In this paper, we discuss canonical number systems (CNSs), which are generalizations of positional number systems to polynomials over the integers. We defined the information quantity of a polynomial A∈Z[x] relative to the base of the CNS and proved that it has a strong relation with the length of the representation in the number system. Based on this result, we showed that for every CNS polynomial P, there exists a finite transducer automaton executing the addition operation of polynomials in canonical representation of base P. Finally, we observed the size—i.e., the number of states—of such automata.

Almost all algebra texts define cyclotomic polynomials using primitive nth roots of unity. However, the elementary formula gcd(xm - 1, xn - 1) = xgcd(m,n) - 1 in ℤ[x] can be used to define the cyclotomic polynomials without reference to roots of unity. In this article, partly motivated by cyclotomic polynomials, we prove a factorization property about strong divisibility sequences in a unique factorization domain. After illustrating this property with cyclotomic polynomials and sequences of the form An − Bn, we use the main theorem to prove that the dynamical analogues of cyclotomic polynomials over any unique factorization domain R are indeed polynomials in R[x]. Furthermore, a converse to this article's main theorem provides a simple necessary and sufficient condition for a divisibility sequence to be a rigid divisibility sequence.

On canonical number systems

Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.