arXiv subjects

Sarah Nakato

Publications and source records attributed to Sarah Nakato.

8 recordsLinked to original sources

arXiv2026

Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$.

math.AC↗

arXiv2024

Primes and absolutely or non-absolutely irreducible elements in atomic domains

We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles.

math.AC↗

arXiv2024

Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely

We study the question up to which power an irreducible integer-valued polynomial that is not absolutely irreducible can factor uniquely. For example, for integer-valued polynomials over principal ideal domains with square-free denominator, already the third power has to factor non-uniquely or the element is absolutely irreducible. Recently, it has been shown that for any $N\in\mathbb{N}$, there exists a discrete valuation domain $D$ and a polynomial $F\in\operatorname{Int}(D)$ such that the minimal $k$ for which $F^k$ factors non-uniquely is greater than $N$. In this paper, we show that, over principal ideal domains with infinitely many maximal ideals of finite index, the minimal power for which an irreducible but not absolutely irreducible element has to factor non-uniquely depends on the $p$-adic valuations of the denominator and cannot be bounded by a constant.

math.AC↗

arXiv2023

Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (\emph{absolutely irreducibles}) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials $F \in \operatorname{Int}(R)$ where $R$ is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number $S\in \mathbb{N}$ that reduces the absolute irreducibility of $F$ to the unique factorization of $F^S$. To this end, we establish a connection between the factors of powers of $F$ and the kernel of a certain linear map that we associate to $F$. This connection yields a characterization of absolute irreducibility in terms of this so-called \emph{fixed divisor kernel}. Given a non-trivial element $\boldsymbol{v}$ of this kernel, we explicitly construct non-trivial factorizations of $F^k$, provided that $k\ge L$, where $L$ depends on $F$ as well as the choice of $\boldsymbol{v}$. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for $k$, one of which only depends on the valuation of the denominator of $F$ and the size of the residue class field of $R$.

math.AC↗

arXiv2022

Split absolutely irreducible integer-valued polynomials over discrete valuation domains

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call \emph{balanced}, characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of $M$. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible.

math.AC↗

arXiv2020

Non-absolutely irreducible elements in the ring of Integer-valued polynomials

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is irreducible in $R$ but for some $n>1$, $r^n$ has other factorizations distinct from $r^n = r \cdots r$, then $r$ is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring $\text{Int}(\mathbb{Z}) = \{f\in \mathbb{Q}[x] \mid f(\mathbb{Z}) \subseteq \mathbb{Z}\}$ of integer-valued polynomials. We also give generalizations of these constructions.

math.AC↗

arXiv2020

A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$, of integer-valued polynomials on a principal ideal domain $D$ with quotient field $K$, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.

math.AC↗

arXiv2019

Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multiplicative monoid of $\operatorname{Int}(D)$ to a block monoid.

math.AC↗