Polynomial Mappings

Narkiewicz, W.

doi:10.1007/bfb0076894

Cited by 43 publications

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“…The fixed divisor has been an object of much recent research, especially in the context where D is a Dedekind domain (see the papers of Bhargava [3,4] or Wood [11]). The monograph of Narkiewicz [9] contains a chapter on this subject, and can be used as a general reference. While Gauss' Lemma indicates that the content behaves nicely under our hypothesis (i.e., c(fg) = c(f )c(g)), the same is not the case for the fixed divisor.…”

Section: On Irreducible Polynomials and Fixed Divisors In Int(s D)mentioning

confidence: 99%

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Chapman

McClain

2005

Journal of Algebra

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Let D be a unique factorization domain and S an infinite subset of D. If f (X) is an element in the ring of integer-valued polynomials over S with respect to D (denoted Int(S, D)), then we characterize the irreducible elements of Int(S, D) in terms of the fixed-divisor of f (X). The characterization allows us to show that every nonzero rational number n/m is the leading coefficient of infinitely many irreducible polynomials in the ring Int(Z) = Int(Z, Z). Further use of the characterization leads to an analysis of the particular factorization properties of such integer-valued polynomial rings. In the case where D = Z, we are able to show that every rational number greater than 1 serves as the elasticity of some polynomial in Int(S, Z) (i.e., Int(S, Z) is fully elastic).

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Section: On Irreducible Polynomials and Fixed Divisors In Int(s D)mentioning

confidence: 99%

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Chapman

McClain

2005

Journal of Algebra

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“…Proposition 3 ( [3], [4], [9], [10]). Let D be an infinite integral domain with quotient field F and unit group U (D).…”

Section: δ-Ringsmentioning

confidence: 99%

δ-rings and factorial sequences preservation

Fares

2006

Acta Arith.

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byYoussef Fares (Amiens) 1. Introduction. For every subset E of Z, and more generally of a Dedekind domain D, Bhargava [2] introduced a notion of generalized factorials associated to E which preserves the classical properties of factorials. Proving a conjecture of Gilmer and Smith [8], we showed in [7] that if a polynomial f ∈ Q[X] maps an infinite subset E of Z onto a subset f (E) which has the same factorials as E, then f is of degree 1. This proof extends easily to the imaginary quadratic number fields but not to all number fields since the group of units may be infinite. The aim of this paper is to give a proof for all number fields. In fact, we prove a stronger result: if a rational function ϕ with coefficients in a number field K transforms an infinite subset E of O K into a subset ϕ(E) which has the same factorials as E, then ϕ is a homographic function (Proposition 26).These questions have strong links with integer-valued polynomials and integer-valued rational functions.Notation. In this paper, D denotes an infinite domain with quotient field F and E denotes a subset of D.Recall [4] that the ring of integer-valued polynomials on E with respect to D isand that the ring of integer-valued rational functions on E with respect to D is the ring

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“…The ring of integer-valued polynomials on an integral domain is well-studied [2] [9]. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules.…”

Section: Introductionmentioning

confidence: 99%

Integer-valued polynomials on commutative rings and modules

Elliott

2017

Communications in Algebra

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Abstract. The ring of integer-valued polynomials on an arbitrary integral domain is well-studied. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring R[T1, . . . , Tn]/(T1 (T1 − r1), . . . , Tn(Tn − rn)) for any commutative ring R and any elements r1, . . . , rn of R, as well as the integer-valued polynomials over the Nagata idealization R(+)M of M over R, where M is an R-module such that every non-zerodivisor on M is a non-zerodivisor of R.

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Polynomial Mappings

Cited by 43 publications

References 0 publications

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

δ-rings and factorial sequences preservation

Integer-valued polynomials on commutative rings and modules

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