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Primary Components of Binomial Ideals
Binomials are polynomials with at most two terms. A binomial ideal is an ideal generated by binomials. Primary components and associated primes of a binomial ideal are still binomial over algebraically closed fields. Primary components of general binomial ideals over algebraically closed fields with characteristic zero can be described combinatorially by translating the operations on binomial ideals to operations on exponent vectors. In this dissertation, we obtain more explicit descriptions for primary components of special binomial ideals. A feature of this work is that our
results are independent of the characteristic of the field.
First of all, we analyze the primary decomposition of a special class of binomial ideals, lattice ideals, in which every variable is a nonzerodivisor modulo the ideal. Then we provide a description for primary decomposition of lattice ideals in fields with positive characteristic.
In addition, we study the codimension two lattice basis ideals and we compute their primary components explicitly.
An ideal I ⊆ k[x_(1),….x_(n) ] is cellular if every variable is either a nonzerodivisor modulo I or is nilpotent modulo I. We characterize the minimal primary components of cellular binomial ideals explicitly. Another significant result is a computation of the Hull of a cellular binomial ideal, that is the intersection of all of its minimal primary components.
Lastly, we focus on commutative monoids and their congruences. We study properties of monoids that have counterparts in the study of binomial ideals. We provide a characterization of primary ideals in positive characteristic, in terms of the congruences they induce
Binomial D-modules
We study quotients of the Weyl algebra by left ideals whose generators
consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators
defined by the grading and a parameter in C^d. We determine the parameters for
which these D-modules (i) are holonomic (equivalently, regular holonomic, when
I is standard-graded); (ii) decompose as direct sums indexed by the primary
components of I; and (iii) have holonomic rank greater than the generic rank.
In each of these three cases, the parameters in question are precisely those
outside of a certain explicitly described affine subspace arrangement in C^d.
In the special case of Horn hypergeometric D-modules, when I is a lattice basis
ideal, we furthermore compute the generic holonomic rank combinatorially and
write down a basis of solutions in terms of associated A-hypergeometric
functions. This study relies fundamentally on the explicit lattice point
description of the primary components of an arbitrary binomial ideal in
characteristic zero, which we derive in our companion article arxiv:0803.3846.Comment: This version is shorter than v2. The material on binomial primary
decomposition has been split off and now appears in its own paper
arxiv:0803.384
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
The primary components of positive critical binomial ideals
A natural candidate for a generating set of the (necessarily prime) defining
ideal of an -dimensional monomial curve, when the ideal is an almost
complete intersection, is a full set of critical binomials. In a somewhat
modified and more tractable context, we prove that, when the exponents are all
positive, critical binomial ideals in our sense are not even unmixed for , whereas for they are unmixed. We further give a complete
description of their isolated primary components as the defining ideals of
monomial curves with coefficients. This answers an open question on the number
of primary components of Herzog-Northcott ideals, which comprise the case
. Moreover, we find an explicit, concrete description of the irredundant
embedded component (for ) and characterize when the hull of the ideal,
i.e., the intersection of its isolated primary components, is prime. Note that
these last results are independent of the characteristic of the ground field.
Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and
Laurent polynomial rings, together with theory of Smith Normal Form and of
Fitting ideals. This gives a more transparent and completely general approach,
replacing the theory of multiplicities used previously to treat the particular
case .Comment: 21 page
Combinatorics of finite abelian groups and Weil representations
The Weil representation of the symplectic group associated to a finite
abelian group of odd order is shown to have a multiplicity-free decomposition.
When the abelian group is p-primary, the irreducible representations occurring
in the Weil representation are parametrized by a partially ordered set which is
independent of p. As p varies, the dimension of the irreducible representation
corresponding to each parameter is shown to be a polynomial in p which is
calculated explicitly. The commuting algebra of the Weil representation has a
basis indexed by another partially ordered set which is independent of p. The
expansions of the projection operators onto the irreducible invariant subspaces
in terms of this basis are calculated. The coefficients are again polynomials
in p. These results remain valid in the more general setting of finitely
generated torsion modules over a Dedekind domain.Comment: 26 pages, 3 figures Revised version, to appear in Pacific Journal of
Mathematic
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