Irreducibility Criterion Research Articles

Monodromy and irreducibility criteria with algorithmic applications in characteristic zero

Monodromy and irreducibility criteria with algorithmic applications in characteristic zero

On an Irreducibility Criterion of M. Ram Murty

(2005). On an Irreducibility Criterion of M. Ram Murty. The American Mathematical Monthly: Vol. 112, No. 3, pp. 269-270.

Monodromy and irreducibility criteria with algorithmic applications in characteristic zero

Consider a projective algebraic variety V, which is the set of all common zeros of homogeneous polynomials of degrees less than d in n + 1 variables over a field of characteristic zero. We suggest an algorithm that decides whether two (or more) given points of V belong to the same irreducible component of V. We also show how to construct, for each s < n, an (s + 1)-dimensional plane in the projective space such that the intersection of every irreducible component of dimension n — s of V with the constructed plane is transversal and is an irreducible curve. These algorithms are deterministic and polynomial in dn and the input size. Bibliography: 9 titles.

Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection − 1 -1 and − 2 -2 , respectively) on a smooth complex projective surface S S which admits a birational morphism π \pi to P 2 \mathbb {P}^{2} . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of C C is in the W W -orbit of the class of the total transform of some point blown up by π \pi if C C is exceptional, or in the W W -orbit of a simple root if C C is root, where W W is the Weyl group acting on Pic ⁡ S \operatorname {Pic}S ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor C C is a necessary and sufficient condition in order that π ∗ ( C ) \pi _{\ast } (C) could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

Towards more accurate separation bounds of empirical polynomials

We study the problem of bounding a polynomial which is absolutely irreducible, away from polynomials which are not absolutely irreducible. These separation bounds are useful for testing whether an empirical polynomial is absolutely irreducible or not, for the given tolerance or error bound of its coefficients. Kaltofen and May studied a method which finds applicable separation bounds using an absolute irreducibility criterion due to Ruppert. In this paper, we study some improvements on their method, by which we are able to find more accurate separation bounds, for bivariate polynomials.

Quasiregular representations of the infinite-dimensional Borel group

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp. 1–156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel group G=Bor 0 N is constructed and a criterion of irreducibility of the constructed representations is presented. This construction uses G-quasi-invariant Gaussian measures on some G-spaces X and extends the method used in Kosyak (Funktsional. Anal. i Priložhen 37 (2003) 78–81) for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group B 0 N .

Structure of Modules Induced from Simple Modules with Minimal Annihilator

AbstractWe study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Imaginary Verma Modules for the Extended Affine Lie Algebra sl 2(ℂ q )

We consider one of the most natural extended affine Lie algebras, the algebra sl 2(ℂ q ), over the quantum torus ℂ q and begin a theory of its representations. In particular, we study a class of imaginary Verma modules, obtain for them a criterion of irreducibility and describe their submodule structure when the modules are in “general position”.

On Mackey's irreducibility criterion for induced representations

Let G be a discrete group, H a subgroup of G, and σ a unitary representation of H. We give an example showing that Mackey's irreducibility criterion for the induced representation IndHG σ does not apply when σ is infinite dimensional.

Irreducibility Criterion for Quasiregular Representations of the Group of Finite Upper Triangular Matrices

An analog of the quasiregular representation is defined for the group of infinite-order finite upper triangular matrices. It uses G-quasi-invariant measures on some G-spaces. The criterion for the irreducibility and equivalence of the constructed representations is given. This criterion allows us to generalize Ismagilov's conjecture on the irreducibility of an analog of regular representations of infinite-dimensional groups.

Representations of graded Hecke algebras

Representations of affine and graded Hecke algebras associated to Weyl groups play an important role in the Langlands correspondence for the admissible representations of a reductive p p -adic group. We work in the general setting of a graded Hecke algebra associated to any real reflection group with arbitrary parameters. In this setting we provide a classification of all irreducible representations of graded Hecke algebras associated to dihedral groups. Dimensions of generalized weight spaces, Langlands parameters, and a Springer-type correspondence are included in the classification. We also give an explicit construction of all irreducible calibrated representations (those possessing a simultaneous eigenbasis for the commutative subalgebra) of a general graded Hecke algebra. While most of the techniques used have appeared previously in various contexts, we include a complete and streamlined exposition of all necessary results, including the Langlands classification of irreducible representations and the irreducibility criterion for principal series representations.

Irreducibility criterion for tensor products of Yangian evaluation modules

The evaluation homomorphisms from the Yangian ${\rm Y}(\mathfrak {gl}\sb n)$ to the universal enveloping algebra ${\rm U}(\mathfrak {gl}\sb n)$ allow one to regard the irreducible finite-dimensional representations of $\mathfrak {gl}\sb n$ as Yangian modules. We give necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.

Difference polynomials and their generalizations

A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefanescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefanescu is a special case.

Irreducibility and Reduction of Discrete-Time Nonlinear Control Systems: An Alternative approach

The problem of irreducibility and system reduction is considered for discrete-time nonlinear single-input single-output systems described by high order input-output (i/o) difference equations. The two seemingly different irreducibility criteria, given in the literature, are shown to be the same except a specific subclass of nonlinear systems. It is suggested how to extend one of those to include the abovementioned subclass. Finally, it is demonstrated how to apply the Maple function univpoly to find the reduced system in case of the polynomial i/o equations.

Discrete valuations centered on local domains

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular, applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-fiberedness, and a valuative criterion for when the normal cone of an ideal in an integrally closed domain is reduced.

Absolute Irreducibility of Polynomials via Newton Polytopes

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero.

Constructing irreducible polynomials with prescribed level curves over finite fields

We use Eisenstein′s irreducibility criterion to prove that there exists an absolutely irreducible polynomial P(X, Y) ∈ GF(q)[X, Y] with coefficients in the finite field GF(q) with q elements, with prescribed level curves Xc : = {(x, y) ∈ GF(q)2 | P(x, y) = c}.

Verma type modules for toroidal lie algebras

We study the structure of imaginary Verma modules induced from the"natural"Borel subalgebra of a toroidal Lie algebra. In particular, we establish a criterion of irreducibility for imaginary Verma modules and describe their submodules and irreducible quotients. We also describe the structure of Verma type modules in the case of sl(2)-toroidal Lie algebra over two variables.

On a generalization of Eisenstein's irreducibility criterion

Let ν be a valuation of any rank of a field K with value group G ν and f(X)= X m + a l X m−1 + … + a m be a polynomial over K. In this paper, it is shown that if (ν(a i )/i)≥(ν(a m )/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(a m )/r∈G ν , then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

Gorenstein Curves with Quasi-Symmetric Weierstrass Semigroups

We consider a canonical Gorenstein curve C of arithmetic genus g in P g-1 (K), that admits a non-singular point P, whose Weierstrass semigroup is quasi-symmetric in the sense that the last gap is equal to 2g-2. By making local considerations at the point P and the second point of the curve C on its osculating hyperplane at P we construct monomial bases for the spaces of higher order regular differentials. We give an irreducibility criterion for the canonical curve in terms of the coefficients of the quadratic relations. We also realize each quasi-symmetric numerical semigroup as the Weierstrass semigroup of a reducible canonical Gorenstein curve, but we give examples of such semigroups that cannot be realized as Weierstrass semigroups of smooth curves.