A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
نویسنده
چکیده
For an arbitrary finite set S of natural numbers greater 1, we construct f ∈ Int(Z)= {g∈Q[x] | g(Z)⊆Z} such that S is the set of lengths of f , i.e., the set of all n such that f has a factorization as a product of n irreducibles in Int(Z). More generally, we can realize any finite multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of f . MSC 2000: primary 13A05, secondary 13B25, 13F20, 20M13, 11C08.
منابع مشابه
Projective P - Orderings and Homogeneous Integer - Valued Polynomials
Bhargava defined p-orderings of subsets of Dedekind domains and with them studied polynomials which take integer values on those subsets. In analogy with this construction for subsets of Z(p) and p-local integer-valued polynomials in one variable, we define projective p-orderings of subsets of Z(p). With such a projective p-ordering for Z(p) we construct a basis for the module of homogeneous, p...
متن کاملRemarks on Polynomial Parametrization of Sets of Integer Points
If, for a subset S of Z, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case), then a ⇒ b (obviously), b ⇒ c, and neither implication is reversible. We...
متن کاملInteger-Valued Polynomials: Looking for Regular Bases (a Survey)
This paper reviews recent results about the additive structure of algebras of integer-valued polynomials, and particularly, the question of the existence and the construction of regular bases. Doing this, we will be led to consider questions of combinatorial, arithmetical, algebraic, ultrametric or dynamical nature. 2010 MSC. Primary 13F20; Secondary 11S05, 11R21, 11B65
متن کاملCountable composition closedness and integer-valued continuous functions in pointfree topology
For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$ of all integer-valued continuous functions, in the usual se...
متن کاملPolynomials with integer coefficients and their zeros
We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for...
متن کامل