1,720,993 research outputs found
Basi di Grobner e Leggi di Raddrizzamento per le bitabelle di Young
No full textThe aim of this paper is to prove that the straightening laws for the letterplace algebra's bideterminants form a Gröbner basis for the ideal generated by Laplace expansions. Since the bideterminants can be interpretated as products of any order minors (subdeterminants) of a matrix of indeterminates, these results extends those obtained by Sturmfels and White who proved that Plucker's identities form a Gröbner basis for the ideal of identities verified only by maximal order minors. The basic ideal of our work is the idea, due to De Concini, Eisenbud and Procesi, that the general case of the minors of any order can be derived from the case of the Plucker's coordinates through a dehomogenizing morphism. In fact, it's well kwown that a Gröbner basis dehomogenizes still in a Gröbner basis provided that certain conditions on term ordering are satisfied
Gröbner bases and gradings for partial difference ideals
No full textIn this paper we introduce a working generalization of the theory of Grobner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear partial difference equations arising, for instance, as discrete models or by the discretization of systems of differential equations. From an algebraic viewpoint, the algebras of partial difference polynomials are free objects in the category of commutative algebras endowed with the action by endomorphisms of a monoid isomorphic to N^r . Then, the investigation of Grobner bases in this context contributes also to the current research trend consisting in studying polynomial rings under the action of suitable symmetries that are compatible with effective methods. Since the algebras of difference polynomials are not Noetherian, we propose in this paper a theory for grading them that provides a Noetherian subalgebra filtration. This implies that the variants of Buchberger’s algorithm we developed for difference ideals terminate in the finitely generated graded case when truncated up to some degree. Moreover, even in the non-graded case, we provide criterions for certifying completeness of eventually finite Grobner bases when they are computed within sufficiently large bounded degrees. We generalize also the concepts of homogenization and saturation, and related algorithms, to the context of difference ideals. The feasibility of the proposed methods is shown by an implementation in Maple that is the first to provide computations for systems of non-linear partial difference equations. We make use of a test set based on the discretization of concrete systems of non-linear partial differential equations
Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms
No full textLet K be the free associative algebra generated by a finite or a countable number of variables x_i . The notion of “letterplace correspondence” introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Grobner bases, J. Symbolic Comput. 44 (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Grobner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K 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 N of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[x_ij , t_j ], where j ranges in N. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Grobner 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
Monomial right ideals and the Hilbert series of noncommutative modules
No full textIn this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module N over the free associative algebra K. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing N 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
Noetherian quotients of the algebra of partial difference polynomials and Grobner bases of symmetric ideals
No full textIn this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain therefore a consistent Grobner bases theory for ideals that possess such symmetries
Strategies for Computing Minimal Free Resolutions
No full textIn the present paper we study algorithms based on the theory of Gröbner bases for
computing free resolutions of modules over polynomial rings. We propose a technique
which consists in the application of special selection strategies to the Schreyer algorithm. The resulting algorithm is efficient and, in the graded case, allows a straightforward minimalization algorithm. These techniques generalize to factor rings, skew commutative rings, and some non-commutative rings. Finally, the proposed approach is compared with other algorithms by means of an implementation developed in the new system Macaulay2
Costandard modules over Schur superalgebras in characteristic
No full textIn this paper we consider the problem of describing the costandard modules ∇(λ) of a
Schur superalgebra S(m|n, r) over a base field K of arbitrary characteristic. Precisely, if
G = GL(m|n) is a general linear supergroup and Dist(G) its distribution superalgebra we
compute the images of the Kostant Z-form under the epimorphism Dist(G) → S(m|n, r).
Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain
an isomorphism ∇(λ) ≈ ∇(λ + |0) ⊗ ∇(0|λ − ) assuming that λ = (λ + |λ − ) and λ m = 0. If char(K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ) p where ∇(μ) is a
costandard module of the ordinary Schur algebra S(n, r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux
Defining relations of low degree of invariants of two 4 x 4 matrices
No full textThe trace algebra C_nd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of C_nd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C_42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple
Inner product functional encryption based on the UOV scheme
No full textWe analyze the efficiency and security of the Inner Product Functional Encryption (IPFE) protocol
introduced in 2021 by Debnath, Mesnager, Dey, and Kundu, specifically when instantiated with UOV.
While the scheme offers several advantages, including improvements in key generation and encryption/decryption
algorithms, along with compact key sizes, the decryption algorithm remains exponential in complexity
with respect to the security parameter. To address this limitation, we propose a variant aimed
at reducing the decryption cost. However, this alternative remains impractical at present due to
the resulting large ciphertext size
Computing minimal free resolutions of right modules over noncommutative algebras
Full text linkIn 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
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