503 research outputs found
Permanental Ideals
The principal result is a primary decomposition of ideals generated by the
(2x2)-subpermanents of a generic matrix. These permanental ideals almost always
have embedded components and their minimal primes are of three distinct
heights. Thus the permanental ideals are almost never Cohen-Macaulay, in
contrast with determinantal ideals.Comment: 13 page
Binomial Ideals and Congruences on Nn
Producción CientÃficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de EconomÃa, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigación FQM-024
Universal gradings of orders
For commutative rings, we introduce the notion of a {\em universal grading},
which can be viewed as the "largest possible grading". While not every
commutative ring (or order) has a universal grading, we prove that every {\em
reduced order} has a universal grading, and this grading is by a {\em finite}
group. Examples of graded orders are provided by group rings of finite abelian
groups over rings of integers in number fields. We generalize known properties
of nilpotents, idempotents, and roots of unity in such group rings to the case
of graded orders; this has applications to cryptography. Lattices play an
important role in this paper; a novel aspect is that our proofs use that the
additive group of any reduced order can in a natural way be equipped with a
lattice structure.Comment: Added section 10; added to and rewrote introduction and abstract (new
Theorem 1.4 and Examples 1.6 and 1.7
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