Unique Factorization Domain Research Articles

A Verified Implementation of Algebraic Numbers in Isabelle/HOL

We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.

Two Properties of Pseudo-Polynomials over a Galois Field

Let F be the completion, with respect to the degree valuation, of the field of rational functions over , the Galois (finite) field of q elements. A function f : F → F is integer-valued if . An integer-valued function f is called a pseudo-polynomial iff(M+K)≡f(M)(modK)for all and . Based on an interpolation series introduced by Carlitzin 1935, explicit shapes of pseudo-polynomials are established. Using an asymptotic characterization of polynomials, it is also proved that the set of all pseudo-polynomials is an integral domain but not a unique factorization domain.

Controlling the dimensions of formal fibers of a unique factorization domain at the height one prime ideals

Let $T$ be a complete local (Noetherian) equidimensional ring with maximal ideal $\mathfrak{m} $ such that the Krull dimension of $T$ is at least two and the depth of $T$ is at least two. Suppose that no integer of $T$ is a zerodivisor and that $|T|=|T/\mathfrak{m} |$. Let $d$ and $t$ be integers such that $1\leq d \leq \dim T-1$, $0 \leq t \leq \dim T - 1$ and $d - 1 \leq t$. Assume that, for every $\mathfrak{p} \in Ass T$, $ht \mathfrak{p} \leq d-1$ and that if $z$ is a regular element of $T$ and $Q \in Ass (T/zT)$, then $ht Q \leq d$. We construct a local unique factorization domain $A$ such that the completion of $A$ is $T$ and such that the dimension of the formal fiber ring at every height one prime ideal of $A$ is $d - 1$ and the dimension of the formal fiber ring of $A$ at $(0)$ is $t$.

On an Equation of Sophie Germain

We deal with the following functional equation f(x)2+4f(y)2=(f(x+y)+f(y))(f(x-y)+f(y))\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(x)^2+4f(y)^2 = \\big ( f(x+y)+f(y) \\big ) \\big ( f(x-y)+f(y) \\big ) \\end{aligned}$$\\end{document}which is motivated by the well known Sophie Germain identity. Some connections as well as some differences between this equation and the quadratic functional equation f(x+y)+f(x-y)=2f(x)+2f(y)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(x+y)+f(x-y)=2f(x)+2f(y) \\end{aligned}$$\\end{document}are exhibited. In particular, the solutions of the quadratic functional equation are expressed in the language of biadditive and symmetric functions, while the solutions of the Sophie Germain functional equation are of the form: the square of an additive function multiplied by some constant. Our main theorem is valid for functions taking values in a unique factorization domain. We present also an example which shows that our main result does not hold in each integral domain.

The Complexity of Primes in Computable Unique Factorization Domains

In many simple integral domains, such as Z or Z[i], there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like Z[x] or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend Z in such a way that we can control the ordinary integer primes in any Π20 way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain (UFD) such that the set of primes is Π20-complete in every computable presentation.

Skolem–Noether algebras

An algebra S is called a Skolem–Noether algebra (SN algebra for short) if for every central simple algebra R, every homomorphism R→R⊗S extends to an inner automorphism of R⊗S. One of the important properties of such an algebra is that each automorphism of a matrix algebra over S is the composition of an inner automorphism with an automorphism of S. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem–Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra S is SN if and only if the power series algebra S[[ξ]] is SN.

Perfect state transfer in unitary Cayley graphs and gcd-graphs

In this work, we consider the problem on the existence of perfect state transfer in unitary Cayley graphs and gcd-graphs over finite commutative rings. We characterize all finite commutative rings allowing perfect transfer to occur on their unitary Cayley graphs. Also, we use our main result to study perfect state transfer in unitary Cayley signed graphs and some gcd-graphs on quotient rings of unique factorization domains. Moreover, we can apply some calculations on the eigenvalues to determine the existence of perfect state transfer in Cayley graphs over finite chain rings.

Reduction of the Small Cohen–Macaulay Conjecture to excellent unique factorization domains

In this article, applying the quasi-Gorenstein analogous of the Ulrich’s deformation of certain Gorenstein rings to unique factorization domains, we show that the Small Cohen–Macaulay Conjecture reduces to excellent unique factorization domains.

Noether's problem on semidirect product groups

Let K be a field, G a finite group. Let G act on the function field L=K(xσ:σ∈G) by τ⋅xσ=xτσ for any σ,τ∈G. Denote the fixed field of the action by K(G)=LG={fg∈L:σ(fg)=fg,∀σ∈G}. Noether's problem asks whether K(G) is rational (purely transcendental) over K.It is known that if G=Cm⋊Cn is a semidirect product of cyclic groups Cm and Cn with Z[ζn] a unique factorization domain, and K contains an eth primitive root of unity, where e is the exponent of G, then K(G) is rational over K.In this paper, we give another criteria to determine whether K(Cm⋊Cn) is rational over K. In particular, if p,q are prime numbers and there exists x∈Z[ζq] such that the norm NQ(ζq)/Q(x)=p, then C(Cp⋊Cq) is rational over C.

Krull dimension and unique factorization in Hurwitz polynomial rings

Let $R$ be a commutative ring with identity, and let $R[x]$ be the collection of polynomials with coefficients in~$R$. We observe that there are many multiplications in $R[x]$ such that, together with the usual addition, $R[x]$ becomes a ring that contains $R$ as a subring. These multiplications belong to a class of functions $\lambda $ from $\mathbb {N}_0$ to $\mathbb {N}$. The trivial case when $\lambda (i) = 1$ for all $i$ gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when $\lambda (i) = i!$ for all $i$. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we study Krull dimension and unique factorization in $R_H[x]$. We show in general that $\dim R \leq \dim R_H[x] \leq 2\dim R +1$. When the ring $R$ is Noetherian we prove that $\dim R \leq \dim R_H[x] \leq \dim R+1$. A condition for the ring $R$ is also given in order to determine whether $\dim R_H[x] = \dim R$ or $\dim R_H[x] = \dim R +1$ in this case. We show that $R_H[x]$ is a unique factorization domain, respectively, a Krull domain, if and only if $R$ is a unique factorization domain, respectively, a Krull domain, containing all of the rational numbers.

Quantum cluster algebra structures on quantum nilpotent algebras

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein–Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts. The proofs rely on Chatters’ notion of noncommutative unique factorization domains. Toric frames are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand–Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the homogeneous prime elements in chains of noncommutative UFDs and their alterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and categorification of cluster algebras. It is expected that this combinatorial model of sequences of homogeneous prime elements will have applications to the unified categorification of quantum nilpotent algebras. Received by the editor 23 December 2013. 2010 Mathematics Subject Classification. Primary 16T20; Secondary 13F60, 17B37, 14M15.

Simple derivations on tensor product of polynomial algebras

Let [Formula: see text] be an unique factorization domain containing a field [Formula: see text] of characteristic zero and let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras. Let [Formula: see text] and [Formula: see text] be two generalized triangular [Formula: see text]-derivations of [Formula: see text] and [Formula: see text], respectively. Denote the unique [Formula: see text]-derivation [Formula: see text] of [Formula: see text] by [Formula: see text]. Then with some conditions on [Formula: see text] and [Formula: see text], it is shown that [Formula: see text] is a simple derivation of [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-simple and [Formula: see text] is [Formula: see text]-simple. We also show that if [Formula: see text] and [Formula: see text] are positively homogeneous derivations and [Formula: see text] is a generalized triangular derivation, then [Formula: see text] is simple derivation of [Formula: see text] if and only if [Formula: see text] is a simple derivation of [Formula: see text] and [Formula: see text] is a simple derivation of [Formula: see text].

A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.

Analogs of Jacobian conditions for subrings

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,…,fm∈k[x1,…,xn], where k is a field of characteristic zero and m∈{1,…,n}. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f1,…,fm]. We also discuss obtained properties in a more general setting – for subrings of unique factorization domains.

An alternating matrix and a vector, with application to Aluffi algebras

Let X be a generic alternating matrix, t be a generic row vector, and J be the ideal Pf4(X)+I1(tX). We prove that J is a perfect Gorenstein ideal of grade equal to the grade of Pf4(X) plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that J defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module N which is equal to the column space of X, calculated mod Pf4(X). The module N is a self-dual maximal Cohen–Macaulay module of rank two; furthermore, J is a Bourbaki ideal for N. The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module N.

Gauss’ lemma and valuation theory

Gauss’ lemma is not only critically important in showing that polynomial rings over unique factorization domains retain unique factorization; it unifies valuation theory. It figures centrally in Krull’s classical construction of valued fields with pre-described value groups, and plays a crucial role in our new short proof of the Ohm-Jaffard-Kaplansky theorem on Bezout domains with given lattice-ordered abelian groups. Furthermore, Eisenstein’s criterion on the irreducibility of polynomials as well as Chao’s beautiful extension of Eisenstein’s criterion over arbitrary domains, in particular over Dedekind domains, are also obvious consequences of Gauss’ lemma. We conclude with a new result which provides a Gauss’ lemma for Hermite rings.

The number of polynomial solutions of polynomial Riccati equations

Consider real or complex polynomial Riccati differential equations a(x)y˙=b0(x)+b1(x)y+b2(x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η+1 (resp. 2) when η≥1 (resp. η=0) and that these bounds are sharp.For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η≥1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η≥2 (resp. η=1) and, again, these bounds are sharp.Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let \(K = \mathbb{Q}\left( {\sqrt d } \right)\), where \(\mathbb{Q}\) is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over \(\mathbb{Q}\). For several quadratic fields \(K = \mathbb{Q}\left( {\sqrt d } \right)\), the ring Rd of integers of K is not a unique-factorization domain. For d was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/ for the cases d = −2,−3.

Primes in quadratic unique factorization domains

The present paper is devoted to extension of a number of well-known results on natural primes for prime elements in quadratic UFD. We obtain analogues of Miller's, Euler's, Lucas' and Pocklington's criterions of primality in quadratic UFD. There is proved that an analogue of the Miller–Rabin test can be realized in quadratic UFD and extended the Rabin result on probability of successful work of the Miller–Rabin test. We construct RSA-cryptosystem in quadratic domains and prove that there hold similar properties to RSA-cryptosystem on integers.

Blowing up finitely supported complete ideals in a regular local ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). We consider singularities of the projective models ProjR[It] and ProjR[It]‾ over SpecR, where R[It]‾ denotes the integral closure of the Rees algebra R[It]. A theorem of Lipman implies that the ideal I has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. If ProjR[It]‾ is regular, we prove that ProjR[It]‾ is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke–Sally in dimension 2, we prove that every local ring S on ProjR[It]‾ that is a unique factorization domain is regular. Moreover, if dim⁡S≥2 and S dominates R, then S is an infinitely near point to R, that is, S is obtained from R by a finite sequence of local quadratic transforms.