A NOTE ON FREE GROUPS IN THE RING OF FRACTIONS OF SKEW POLYNOMIAL RINGS

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t; sigma], the skew polynomial ring in t, over L, with automorphism sigma. We prove that the multiplicative group D(x) of D contains a free noncyclic subgroup.

A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.

Encoding through generalized polynomial codes

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

A SURVEY ON FREE OBJECTS IN DIVISION RINGS AND IN DIVISION RINGS WITH AN INVOLUTION

Let D be a division ring with center k, and let D-dagger be its multiplicative group. We investigate the existence of free groups in D-dagger, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D-dagger.

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a unital Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent to a bounded complex of finitely generated projective L0-modules, generalising known results for twisted Laurent polynomial rings. The crucial hypothesis is that L is a strongly graded ring. 

GLOBALIZATION OF TWISTED PARTIAL ACTIONS

Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish a criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.

Mechanism and model of the oscillatory electro-oxidation of methanol

A mechanism for the kinetic instabilities observed in the galvanostatic electro-oxidation of methanol is suggested and a model developed. The model is investigated using stoichiometric network analysis as well as concepts from algebraic geometry (polynomial rings and ideal theory) revealing the occurrence of a Hopf and a saddle-node bifurcation. These analytical solutions are confirmed by numerical integration of the system of differential equations. (C) 2010 American Institute of Physics

Métodos eficientes para criptografia baseada em reticulados.

Reticulados têm sido aplicados de diferentes maneiras em criptografia. Inicialmente utilizados para a destruição de criptossistemas, eles foram posteriormente aplicados na construção de novos esquemas, incluindo criptossistemas assimétricos, esquemas de assinatura cega e os primeiros métodos para encriptação completamente homomórfica. Contudo, seu desempenho ainda é proibitivamente lenta em muitos casos. Neste trabalho, expandimos técnicas originalmente desenvolvidas para encriptação homomórfica, tornando-as mais genéricas e aplicando-as no esquema GGH-YK-M, um esquema de encriptação de chave pública, e no esquema LMSV, a única construção homomórfica que não sucumbiu a ataques de recuperação de chaves IND-CCA1 até o momento. Em nossos testes, reduzimos o tamanho das chaves do GGH-YK-M em uma ordem de complexidade, especificamente, de O(n2 lg n) para O(n lg n), onde n é um parâmetro público do esquema. A nova técnica também atinge processamento mais rápido em todas as operações envolvidas em um criptossistema assimétrico, isto é, geração de chaves, encriptação e decriptação. A melhora mais significativa é na geração de chaves, que se torna mais de 3 ordens de magnitude mais rápida que resultados anteriores, enquanto a encriptação se torna por volta de 2 ordens de magnitude mais rápida. Para decriptação, nossa implementação é dez vezes mais rápida que a literatura. Também mostramos que é possível aumentar a segurança do esquema LMSV contra os ataques quânticos de recuperação de chaves recentemente publicados pela agência britânica GCHQ. Isso é feito através da adoção de reticulados não-ciclotômicos baseados em anéis polinomiais irredutíveis quase-circulantes. Em nossa implementação, o desempenho da encriptação é virtualmente idêntico, e a decriptação torna-se ligeiramente inferior, um pequeno preço a se pagar pelo aumento de segurança. A geração de chaves, porém, é muito mais lenta, devido à necessidade de se utilizar um método mais genérico e caro. A existência de métodos dedicados altamente eficientes para a geração de chaves nesta variante mais segura do LMSV permanece como um problema em aberto.

On decomposition of sub-linearised polynomials

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

Cyclic Maximal Ideals of Rings of Differential Operators over Power Series Rings

In [3], Bratti and Takagi conjectured that a first order differential operator S=11 +...+ nn+ with 1,..., n, {x1,..., xn} does not generate a cyclic maximal left (or right) ideal of the ring of differential operators. This is contrary to the case of the Weyl algebra, i.e., the ring of differential operators over the polynomial ring [x1,..., xn]. In this case, we know that such cyclic maximal ideals do exist. In this article, we prove several special cases of the conjecture of Bratti and Takagi.

POLYNOMIAL AND GROUP IDENTITIES IN ALTERNATIVE LOOP ALGEBRAS

We investigate polynomial identities on an alternative loop algebra and group identities on its (Moufang) unit loop. An alternative loop ring always satisfies a polynomial identity, whereas whether or not a unit loop satisfies a group identity depends on factors such as characteristic and centrality of certain kinds of idempotents.

On invariant Rings of Sylow subgroups of finite classical groups

In this thesis we study the invariant rings for the Sylow p-subgroups of the nite classical groups. We have successfully constructed presentations for the invariant rings for the Sylow p-subgroups of the unitary groups GU(3; Fq2) and GU(4; Fq2 ), the symplectic group Sp(4; Fq) and the orthogonal group O+(4; Fq) with q odd. In all cases, we obtained a minimal generating set which is also a SAGBI basis. Moreover, we computed the relations among the generators and showed that the invariant ring for these groups are a complete intersection. This shows that, even though the invariant rings of the Sylow p-subgroups of the general linear group are polynomial, the same is not true for Sylow p-subgroups of general classical groups. We also constructed the generators for the invariant elds for the Sylow p-subgroups of GU(n; Fq2 ), Sp(2n; Fq), O+(2n; Fq), O-(2n + 2; Fq) and O(2n + 1; Fq), for every n and q. This is an important step in order to obtain the generators and relations for the invariant rings of all these groups.

Digit systems over commutative rings

Let epsilon be a commutative ring with identity and P is an element of epsilon[x] be a polynomial. In the present paper we consider digit representations in the residue class ring epsilon[x]/(P). In particular, we are interested in the question whether each A is an element of epsilon[x]/(P) can be represented modulo P in the form e(0)+ e(1)x + ... + e(h)x(h), where the e(i) is an element of epsilon[x]/(P) are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

A Note on Linear Codes over Semigroup Rings

In this paper, we introduced new construction techniques of BCH, alternant, Goppa, Srivastava codes through the semigroup ring B[X; 1 3Z0] instead of the polynomial ring B[X; Z0], where B is a finite commutative ring with identity, and for these constructions we improve the several results of [1]. After this, we present a decoding principle for BCH, alternant and Goppa codes which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight t ≤ r/2, i.e., whose minimum Hamming distance is r + 1.