Unique Factorization Research Articles

Factorizations in evaluation monoids of Laurent semirings

For let be the semiring of real numbers with all where is the set of nonnegative integers and is the semiring of Laurent polynomials with coefficients in In this paper, we study various factorization properties of the additive structure of We characterize when is atomic. Then we characterize when satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when satisfies the unique factorization property and show that, when this is not the case, has infinite elasticity.

Factorization Patterns in F<sub><i>q</i></sub> [<i>x</i>

The finite field Fq has q elements, where q = pk for prime p and k∈N. Then Fq[x] is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f(x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥deg c. Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f(x) is itself irreducible over Fq, it would belong to the template [n], and if f(x) split over Fq, it would belong to the template [n] Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of Fq[x] belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors.

Tensor product decompositions and rigidity of full factors

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g. $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

Completions of uncountable local rings with countable spectra

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results provide a way to construct uncountable local domains with countable spectra from a much broader class of complete local rings than was previously known. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.

Generalizations of Samuel's criteria for a ring to be a unique factorization domain

We give several criteria for a ring to be a UFD, including generalizations of some criteria due to P. Samuel. These criteria are applied to construct, for any field k, (1) a Z-graded non-noetherian rational UFD of dimension 3 over k, and (2) k-affine rational UFDs defined by trinomial relations.

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

On the arithmetic of power monoids and sumsets in cyclic groups

Let $H$ be a multiplicatively written monoid with identity $1_H$ (in particular, a group). We denote by $\mathcal P_{\rm fin,\times}(H)$ the monoid obtained by endowing the collection of all finite subsets of $H$ containing a unit with the operation of setwise multiplication $(X,Y) \mapsto \{xy: x \in X, y \in Y\}$; and study fundamental features of the arithmetic of this and related structures, with a focus on the submonoid, $\mathcal P_{\text{fin},1}(H)$, of $\mathcal P_{\text{fin},\times}(H)$ consisting of all finite subsets $X$ of $H$ with $1_H \in X$. Among others, we prove that $\mathcal{P}_{\text{fin},1}(H)$ is atomic (i.e., each non-unit is a product of irreducibles) iff $1_H \ne x^2 \ne x$ for every $x \in H \setminus \{1_H\}$. Then we obtain that $\mathcal{P}_{\text{fin},1}(H)$ is BF (i.e., it is atomic and every element has factorizations of bounded length) iff $H$ is torsion-free; and show how to transfer these conclusions to $\mathcal P_{\text{fin},\times}(H)$. Next, we introduce "minimal factorizations" to account for the fact that monoids may have non-trivial idempotents, in which case standard definitions from Factorization Theory degenerate. Accordingly, we obtain conditions for $\mathcal P_{\text{fin},\times}(H)$ to be BmF (meaning that each non-unit has minimal factorizations of bounded length); and for $\mathcal{P}_{\text{fin},1}(H)$ to be BmF, HmF (i.e., a BmF-monoid where all the minimal factorizations of a given element have the same length), or minimally factorial (i.e., a BmF-monoid where each element has an essentially unique minimal factorization). Finally, we prove how to realize certain intervals as sets of minimal lengths in $\mathcal P_{\text{fin},1}(H)$. Many proofs come down to considering sumset decompositions in cyclic groups, so giving rise to an intriguing interplay with Arithmetic Combinatorics.

Wave Transport and Localization in Prime Number Landscapes

In this paper, we study the wave transport and localization properties of novel aperiodic structures that manifest the intrinsic complexity of prime number distributions in imaginary quadratic fields. In particular, we address structure-property relationships and wave scattering through the prime elements of the nine imaginary quadratic fields (i.e., of their associated rings of integers) with class number one, which are unique factorization domains (UFDs). Our theoretical analysis combines the rigorous Green’s matrix solution of the multiple scattering problem with the interdisciplinary methods of spatial statistics and graph theory analysis of point patterns to unveil the relevant structural properties that produce wave localization effects. The onset of a Delocalization-Localization Transition (DLT) is demonstrated by a comprehensive study of the spectral properties of the Green’s matrix and the Thouless number as a function of their optical density. Furthermore, we employ Multifractal Detrended Fluctuation Analysis (MDFA) to establish the multifractal scaling of the local density of states in these complex structures and we discover a direct connection between localization, multifractality, and graph connectivity properties. Finally, we use a semi-classical approach to demonstrate and characterize the strong coupling regime of quantum emitters embedded in these novel aperiodic environments. Our study provides access to engineering design rules for the fabrication of novel and more efficient classical and quantum sources as well as photonic devices with enhanced light-matter interaction based on the intrinsic structural complexity of prime numbers in algebraic fields.

New results on Nyldon words and Nyldon-like sets

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with “nondecreasing” replaced by “nonincreasing.” Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier, Philibert, and Stipulanti. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name “Nyldon-like sets”, and show how to generate these sets easily.

Mustafin Models of Projective Varieties and Vector Bundles

Abstract Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

Distinguishing L-functions by joint universality

In this note, we present results for distinguishing L-functions by their multisets of zeros and unique factorizations in an axiomatic setting; our tools stem from universality theory.

Computation of the suffix array, Burrows-Wheeler transform and FM-index in V-order

V-order is a total order on strings that determines an instance of Unique Maximal Factorization Families (UMFFs), a generalization of Lyndon words. The fundamental V-comparison of strings can be done in linear time and constant space. V-order has been proposed as an alternative to lexicographic order (lexorder) in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform (BWT). In line with the recent interest in the connection between suffix arrays and Lyndon factorization, in this paper we obtain similar results for the V-order factorization. Indeed, we show that the results describing the connection between suffix arrays and Lyndon factorization are matched by analogous V-order processing. We also describe a methodology for efficiently computing the FM-Index in V-order, as well as V-order substring pattern matching using backward search.

A Quick Route to Unique Factorization in Quadratic Orders

We give a short proof—not relying on ideal classes or the geometry of numbers—of a known criterion for quadratic orders to possess unique factorization.

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.

Algorithms for linear time reconstruction by discrete tomography II

The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a field or a unique factorization domain, such as R or Z. We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.

Subrings of the power series ring over a principal ideal domain

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series in such that for all large i. By definition, the polynomial ring and the power series ring are special cases of when and I = D, respectively. In this article, we investigate the ring in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS , where I = dD with and Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.

Unique factorization properties in commutative monoid rings with zero divisors

Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of “unique factorization rings.” Our work generalizes Anderson et al.’s results about “unique factorization” in R[X], Gilmer and Parker’s characterization of factorial monoid domains, and Hardy and Shores’s classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is “restricted cancellative” or satisfies various “(restricted) ideal cancellation laws.”

Length-factoriality in commutative monoids and integral domains

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x∈M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is, in general, weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M, and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely long and purely short irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

Efficient q-integer linear decomposition of multivariate polynomials

We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and for describing the q-counterpart of Ore-Sato theory. Both of our algorithms require only basic integer and polynomial arithmetic and work for any unique factorization domain containing the ring of integers. Complete complexity analyses are conducted for both our algorithms and two previous algorithms in the case of multivariate integer polynomials, showing that our algorithms have better theoretical performances. A Maple implementation is also included which suggests that our algorithms are much faster in practice than previous algorithms.

Direct product primality testing of graphs is GI-hard

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.