100 research outputs found
Computing minimal free resolutions of right modules over noncommutative algebras
In this paper we propose a general method for computing a minimal free right
resolution of a finitely presented graded right module over a finitely
presented graded noncommutative algebra. In particular, if such module is the
base field of the algebra then one obtains its graded homology. The approach is
based on the possibility to obtain the resolution via the computation of
syzygies for modules over commutative algebras. The method behaves
algorithmically if one bounds the degree of the required elements in the
resolution. Of course, this implies a complete computation when the resolution
is a finite one. Finally, for a monomial right module over a monomial algebra
we provide a bound for the degrees of the non-zero Betti numbers of any single
homological degree in terms of the maximal degree of the monomial relations of
the module and the algebra.Comment: 23 pages, to appear in Journal of Algebr
Monomial right ideals and the Hilbert series of noncommutative modules
In this paper we present a procedure for computing the rational sum of the
Hilbert series of a finitely generated monomial right module over the free
associative algebra . We show that such
procedure terminates, that is, the rational sum exists, when all the cyclic
submodules decomposing are annihilated by monomial right ideals whose
monomials define regular formal languages. The method is based on the iterative
application of the colon right ideal operation to monomial ideals which are
given by an eventual infinite basis. By using automata theory, we prove that
the number of these iterations is a minimal one. In fact, we have experimented
efficient computations with an implementation of the procedure in Maple which
is the first general one for noncommutative Hilbert series.Comment: 15 pages, to appear in Journal of Symbolic Computatio
Multigraded Hilbert Series of noncommutative modules
In this paper, we propose methods for computing the Hilbert series of
multigraded right modules over the free associative algebra. In particular, we
compute such series for noncommutative multigraded algebras. Using results from
the theory of regular languages, we provide conditions when the methods are
effective and hence the sum of the Hilbert series is a rational function.
Moreover, a characterization of finite-dimensional algebras is obtained in
terms of the nilpotency of a key matrix involved in the computations. Using
this result, efficient variants of the methods are also developed for the
computation of Hilbert series of truncated infinite-dimensional algebras whose
(non-truncated) Hilbert series may not be rational functions. We consider some
applications of the computation of multigraded Hilbert series to algebras that
are invariant under the action of the general linear group. In fact, in this
case such series are symmetric functions which can be decomposed in terms of
Schur functions. Finally, we present an efficient and complete implementation
of (standard) graded and multigraded Hilbert series that has been developed in
the kernel of the computer algebra system Singular. A large set of tests
provides a comprehensive experimentation for the proposed algorithms and their
implementations.Comment: 28 pages, to appear in Journal of Algebr
Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms
Let be the free associative algebra generated by a finite or
countable number of variables . The notion of "letterplace correspondence"
introduced in [1,2] for the graded (two-sided) ideals of is
extended in this paper also to the nongraded case. This amounts to the
possibility of modelizing nongraded noncommutative presented algebras by means
of a class of graded commutative algebras that are invariant under the action
of the monoid of natural numbers. For such purpose we develop the
notion of saturation for the graded ideals of , where is an
extra variable and for their letterplace analogues in the commutative
polynomial algebra , where ranges in . In
particular, one obtains an alternative algorithm for computing inhomogeneous
noncommutative Gr\"obner bases using just homogeneous commutative polynomials.
The feasibility of the proposed methods is shown by an experimental
implementation developed in the computer algebra system Maple and by using
standard routines for the Buchberger algorithm contained in Singular.
References
[1] La Scala, R.; Levandovskyy, V., Letterplace ideals and non-commutative
Gr\"obner bases. J. Symbolic Comput., 44 (2009), 1374--1393.
[2] La Scala, R.; Levandovskyy, V., Skew polynomial rings, Gr\"obner bases
and the letterplace embedding of the free associative algebra. J. Symbolic
Comput., 48 (2013), 110--131Comment: 22 pages, to appear in International Journal of Algebra and
Computatio
Weak Polynomial Identities for M1,1(E)
* Partially supported by Universita` di Bari: progetto “Strutture algebriche, geometriche e descrizione degli invarianti ad esse associate”.We compute the cocharacter sequence and generators of the
ideal of the weak polynomial identities of the superalgebra M1,1 (E)
Stream/block ciphers, difference equations and algebraic attacks
In this paper we introduce a general class of stream and block ciphers that
are defined by means of systems of (ordinary) explicit difference equations
over a finite field. We call this class "difference ciphers". Many important
ciphers such as systems of LFSRs, Trivium/Bivium and Keeloq are difference
ciphers. To the purpose of studying their underlying explicit difference
systems, we introduce key notions as state transition endomorphisms and show
conditions for their invertibility. Reducible and periodic systems are also
considered. We then propose general algebraic attacks to difference ciphers
which are experimented by means of Bivium and Keeloq.Comment: 22 page
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