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Register a new userIrreducibility of integer-valued polynomials I
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Authors
Devendra Prasad
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Let $S subset R$ be an arbitrary subset of a unique factorization domain $R$ and $K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $mathrm{Int}(S,R)= { f in mathbb{K}[x]: f(a) in R forall a in S }.$ This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call $d$-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in $mathrm{Int}(S,R).$ In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in $mathrm{Int}(S,R).$ At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.
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In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.
Using the following $_4F_3$ transformation formula $$ sum_{k=0}^{n}{-x-1choose k}^2{xchoose n-k}^2=sum_{k=0}^{n}{n+kchoose 2k}{2kchoose k}^2{x+kchoose 2k}, $$ which can be proved by Zeilbergers algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on integer-valued polynomials.
We consider the question of certifying that a polynomial in ${mathbb Z}[x]$ or ${mathbb Q}[x]$ is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv.~that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in ${mathbb F}_p[x]$ are self-certifying.
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $fcolon Gto E$. A function $fin C(G,E)$ is a generalized polynomial, if there is an $nge 0$ such that $Delta_{h_1} ldots Delta_{h_{n+1}} f=0$ for every $h_1 ,ldots , h_{n+1} in G$, where $Delta_h$ is the difference operator. We say that $fin C(G,E)$ is a polynomial, if it is a generalized polynomial, and the linear span of its translates is of finite dimension; $f$ is a w-polynomial, if $ucirc f$ is a polynomial for every $uin E^*$, and $f$ is a local polynomial, if it is a polynomial on every finitely generated subsemigroup. We show that each of the classes of polynomials, w-polynomials, generalized polynomials, local polynomials is contained in the next class. If $G$ is an Abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. We introduce the classes of exponential polynomials and w-expo-nential polynomials as well, establish their representations and connection with polynomials and w-polynomials. We also investigate spectral synthesis and analysis in the class $C(G,E)$. It is known that if $G$ is a compact Abelian group and $E$ is a Banach space, then spectral synthesis holds in $C(G,E)$. On the other hand, we show that if $G$ is an infinite and discrete Abelian group and $E$ is a Banach space of infinite dimension, then even spectral analysis fails in $C(G,E)$. If, however, $G$ is discrete, has finite torsion free rank and if $E$ is a Banach space of finite dimension, then spectral synthesis holds in $C(G,E)$.
We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > operatorname{deg}(c) + operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $mathbb Q$, where $c, d in mathbb Z[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N ge 5$ and $k in {3, 4, ldots, 24} setminus {9, 16}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} pm (l x + 1)$ with $k, l in mathbb Z$, $k eq l$.
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