Integer-valued skew polynomials

Given an integral domain [Formula: see text] with quotient field [Formula: see text], the study of the ring of integer-valued polynomials [Formula: see text] has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if [Formula: see text] is an automorphism of [Formula: see text], one may consider the set [Formula: see text], where [Formula: see text] is the skew polynomial ring and [Formula: see text] is a “suitable” evaluation of [Formula: see text] at [Formula: see text]. For example, he gave sufficient conditions for [Formula: see text] to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring [Formula: see text] with a suitable evaluation, where [Formula: see text] is a [Formula: see text]-derivation. Moreover we prove, for example, that if [Formula: see text] is of finite order and [Formula: see text] is a Dedekind domain with finite residue fields such that [Formula: see text] is a ring, then [Formula: see text] is non-Noetherian.

Let R be a commutative ring with identity. An element is said to be absolutely irreducible in R if for all natural numbers n > 1, rn has essentially only one factorization namely If is irreducible in R but for some n > 1, rn has other factorizations distinct from then r is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring of integer-valued polynomials. We also give generalizations of these constructions.

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

ABSTRACTThe 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 for any commutative ring R and any elements 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. These examples are motivated by and provide motivation for the study of integer-valued derivatives.

Let (R,m) be a complete discrete valuation ring with the finite residue field R∕m= Fq. Given a monic polynomial P(t)∈R[t] whose reduction modulo m gives an irreducible polynomial P‾(t)∈Fq[t], we initiate an investigation of the distribution of coker(P(A)), where A∈Matn(R) is randomly chosen with respect to the Haar probability measure on the additive group Matn(R) of n×n R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between Matn(R) and Matn(Fq) to translate our problems about a Haar-random matrix in Matn(R) into problems about a random matrix in Matn(Fq) with respect to the uniform distribution. Our results over Fq are about the distribution of the P‾-part of a random matrix A‾∈Matn(Fq) with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., Zp-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of Q with a probability inversely proportional to |AutZ(H)|. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

We introduce two new notions of “ P P -ordering” and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of P P -orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) P P -adic analysis. Specifically, we first use these notions of P P -orderings and factorials to construct explicit Pólya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain. Second, we classify “smooth” functions on an arbitrary compact subset S S of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on S S satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler’s Theorem (classifying the functions that are continuous on Z p \mathbb {Z}_p ) to a very general setting. In particular, our constructions prove that, for any ϵ > 0 \epsilon >0 , the functions in any of the above Banach spaces can be ϵ \epsilon -approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallée-Poussin, and Bernstein. Our proofs are effective.

The fundamental result of linear network coding asserts the existence of optimal codes over acyclic networks when the symbol field is sufficiently large. The restriction to just acyclic networks turns out to stem from the customary algebraic structure of data symbols as a finite field. Adopting data units that belong to a discrete valuation ring (DVR), that is, a PID with a unique maximal ideal, much of the network coding theory extends to cyclic networks. Being a PID with the maximal ideal 0, a field can be regarded as a degenerated DVR. Thus the field-based theory becomes a degenerated version of the DVR-based theory. Meanwhile, convolutional network coding becomes the instance when the DVR consists of rational power series over a field. Besides the treatise in commutative algebra, the present paper also delves into the efficiency issue of code construction. Given a cyclic network, a quadratically large acyclic network is constructed so that every optimal code on the acyclic network subject to some straightforward restriction induces an optimal code on the given network. In this way, existing construction algorithms over acyclic networks can be adapted for cyclic networks as well.

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

The ring of integer-valued polynomials over a given subset S of ℤ (or Int(S,ℤ)) is defined as the set of polynomials in ℚ[x] which maps S to ℤ. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial f∈Int(S,ℤ)). We also generalize our results to arbitrary subsets of a Dedekind domain.

A result of Farahat and Higman shows that there is a “universal” algebra, FH , interpolating the centres of symmetric group algebras, Z ( ℤ S n ) . We explain that this algebra is isomorphic to ℛ ⊗ Λ , where ℛ is the ring of integer-valued polynomials and Λ is the ring of symmetric functions. Moreover, the isomorphism is via “evaluation at Jucys–Murphy elements”, which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products Γ ≀ S n of a fixed finite group Γ . This involves constructing wreath-product versions ℛ Γ and Λ ( Γ * ) of ℛ and Λ , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, FH Γ , is isomorphic to ℛ Γ ⊗ Λ ( Γ * ) and use this to compute the p -blocks of wreath products.

Generating ideals in rings of integer-valued polynomials

Let D be a Dedekind domain with quotient field K. The ring of integer-valued polynomials on D is the subring Int(D) = { f ∈ K[X]: f(D) ⊆ D} of the polynomial ring K[X]. The Polya-Ostrowski group PO(D) of D is a subgroup of the class group of D generated by the well-known factorial ideals n! D of D. A regular basis of Int(D) is a D-module basis consisting of one polynomial of each degree. It is well known that Int(D) has a regular basis if and only if the group PO(D) is trivial, if and only if the D-module Int n (D) = { f ∈ Int(D): degf ≤ n} is free for all n. In this paper we provide evidence for and prove special cases of the conjecture that, if PO(D) is finite, then the natural density of the set of nonnegative integers n such that Int n (D) is free exists, is rational, and is at least 1∕ | PO(D) | . Moreover, we compute this density or determine a conjectural value for several examples of Galois number fields of degrees 2, 3, 4, 5, and 6 over \(\mathbb{Q}\).

Another Prüfer Ring of Integer-Valued Polynomials

Let [Formula: see text] be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal [Formula: see text] of [Formula: see text], the residue field [Formula: see text] is finite. Let [Formula: see text] be the quotient field of [Formula: see text]. We investigate sets of lengths in the ring of integer-valued polynomials on [Formula: see text], [Formula: see text]M, D[Formula: see text]. For every multiset of integers [Formula: see text], we explicitly construct an element of Int([Formula: see text]) with exactly [Formula: see text] essentially different factorizations into irreducible elements of Int([Formula: see text] whose lengths are [Formula: see text]. Furthermore, we show that Int([Formula: see text]) is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings Int([Formula: see text]) ≠Int([Formula: see text]), where [Formula: see text] is an infinite subset of [Formula: see text].

A submonoid S of a given monoid H is called monadic if it is a divisor-closed submonoid of H generated by one element (i.e., there is some (non-zero) b ∈ H such that S is the smallest divisor-closed submonoid of H such that b ∈ S). In this paper we study monoids and domains whose monadic submonoids are Krull monoids. These monoids resp. domains are called monadically Krull. Every Krull monoid is a monadically Krull monoid, but the converse is not true. We provide several types of counterexamples and present a few characterizations for monadically Krull monoids. Furthermore, we show that rings of integer-valued polynomials over factorial domains are monadically Krull. Finally, we investigate the connections between monadically Krull monoids and generalizations of SP-domains.