Irreducibility Criterion Research Articles

Polynomials in Base $x$ and the Prime-Irreducible Affinty

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique “base-$x$” representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.

Criteria of irreducibility of the Koopman representations for the group [formula omitted

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Problem 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group GL0(2∞,R)=lim→nGL(2n−1,R), the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian non-centered measures. The corresponding G-space Xm is a subspace of the space Mat(2∞,R) of infinite in both directions real matrices. In fact, Xm is a collection of m infinite in both directions rows. This result was announced in [22]. We give the proof only for m≤2. The general case will be studied later.

C*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ϕ on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l2(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.

Irreducibility criteria for compositions of multivariate polynomials

We provide irreducibility criteria for compositions of multivariate polynomials over a field K, of the form \({f(X_{1},\ldots ,X_{r-1},g(X_{1},\ldots ,X_{r}))}\), with \({f,g\in K[X_{1},\ldots ,X_{r}]}\), for the case that \({f}\) as a polynomial in X r is irreducible over \({K(X_{1},\ldots ,X_{r-1})}\) and has leading coefficient divisible by a power of an irreducible polynomial \({p(X_{1},\ldots ,X_{r-1})}\) of sufficiently large degree with respect to \({X_{r-1}}\).

C*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ϕ on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l2(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.

Algebraic degrees of pseudo-Anosov stretch factors

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichm\"uller space of $S$. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anosov stretch factors on almost all finite type surfaces. As a corollary, we find the possible degrees of the number fields that arise as trace fields of Veech groups of flat surfaces homeomorphic to closed orientable surfaces. Our construction also gives an algorithm for finding a pseudo-Anosov map on a given surface whose stretch factor has a prescribed degree. One ingredient of the proofs is a novel asymptotic irreducibility criterion for polynomials.

Determinant formula for parabolic Verma modules of Lie superalgebras

We prove a determinant formula for a parabolic Verma module of a contragredient finite-dimensional Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module.

ALGORITHMS FOR FACTORIZATION AND IRREDUCIBILITY TESTING OF POLYNOMIALS USING ELLIPTIC CURVES

In this article irreducibility criteria and factorization algorithms are formulated and proved, that are generalizations of correspondent criteria and algorithms for integers. Generalization of Lenstra’s theorem is of especial interest. It provides correct calculations of multiple points on elliptic curve that is used in Lenstra algorithm. Note that the generalization of this theorem in case of finite field of characteristic 2 is completely different from the classical case and is very non-trivial. Using the criteria obtained, we construct a series of algorithms for factoring and testing of irreducibility of polynomials over finite field. We should note that their performance isn’t their main advantage, but the main advantage is using some abelian group built on elliptic curve. So in case of failure of algorithm we can just chose the other elliptic curve over the corresponding field and repeat the algorithm. These algorithms have no large advantage in speed, like their preimages – analogical algorithms for integers, but they demonstrate new approaches to the solutions of important problem of irreducibility testing and also are very interesting from mathematical point of view.

A result on the equation $x^p + y^p = z^r$ using Frey abelian varieties

We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields.

An irreducibility criterion for power series

We prove an irreducibility criterion for polynomials with power series coefficients generalizing previous results given by García Barroso and González-Pérez.

On a mild generalization of the Schönemann - Eisenstein - Dumas irreducibility criterion

ABSTRACTWe state a mild generalization of the classical Schönemann and Eisenstein- Dumas irreducibility criterion in ℤ[x] and provide an elementary proof. In the end of the paper, we also provide a concrete example of a polynomial which is irreducible by the main result of the paper but whose irreducibility does not follow from existing criteria.

Irreducible admissible mod-p representations of metaplectic groups

Let p be an odd prime number, and F a nonarchimedean local field of residual characteristic p. We classify the smooth, irreducible, admissible genuine mod-p representations of the twofold metaplectic cover $$\widetilde{\text {Sp}}_{2n}(F)$$ of $$\text {Sp}_{2n}(F)$$ in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart–Vigneras as well as new technical results to adapt Herzig’s method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations generalizing the complete irreducibility of principal series representations in the rank 1 case, as well as the fact that irreducibility is preserved by parabolic induction from the cover of the Siegel Levi subgroup.

A Generalization of the Eisenstein–Dumas–Schönemann Irreducibility Criterion

AbstractIn 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

An irreducibility criterion for power series

An irreducibility criterion for power series

Irreducibility Criteria for Reciprocal Polynomials and Applications

We present criteria for determining irreducibility of reciprocal polynomials over the field of rational numbers. We also obtain some combinatorial results concerning the irreducibility of reciprocal polynomials. As a consequence of our approach, we are able to deal with other problems such as factorization properties of Chebyshev polynomials of the first and second kind and with the classical problems of computing minimal polynomials of algebraic values of trigonometric functions.

On a mild generalization of the Schönemann irreducibility criterion

ABSTRACTWe state a mild generalization of the classical Schönemann irreducibility criterion in ℤ[x] and provide an elementary proof.

An irreducibility criterion for the sum of two relatively prime polynomials

We extend a result of Cavachi on sums of relatively prime polynomials by proving that a polynomial of the form $f(X)+p^{k}g(X)$, with $f$ and $g$ relatively prime polynomials with integer coefficients, $\deg f<\deg g$, and $k$ a positive integer prime to $\deg g$ is irreducible over $\mathbb{Q}$ for all but finitely many prime numbers $p$.

Irreducibility criteria for compositions of polynomials with integer coefficients

We provide irreducibility criteria for some classes of compositions of polynomials with integer coefficients of the form $$F\circ G$$ , with F being a quadratic irreducible polynomial and G a polynomial of arbitrary degree.

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.

Further irreducibility criteria for polynomials with non-negative coefficients

Further irreducibility criteria for polynomials with non-negative coefficients