اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDivisibility in rings of integer-valued polynomials
344
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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.
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 con
jecture of Z.-W. Sun on integer-valued polynomials.
In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tig
ht Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a qua
dratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{lfloor n/2rfloor}$ divides $sum_{piinmathfrak{S}_n}q^{rm{altmaj}(pi)}$, where $mathfrak{S}_n$ is the set of all permutations of ${1,2,ldots,n}$ and $rm{altmaj}(pi)$ is the alternating major index of $pi$. This leads us to discover a $q$-analog of $n!=2^{ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $gamma$-vectors of the alternating descent polynomials by using these two recursions and the ${textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
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 $C(G,E)$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
