Initial Ideals Research Articles

Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian

A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).

Generic circuits sets and general initial ideals with respect to weights

We study the set of circuits of a homogeneous ideal and that of its truncations, and introduce the notion of generic circuits set. We show how this is a well-defined in- variant that can be used, in the case of initial ideals with respect to weights as a counterpart of the (usual) generic ini- tial ideal with respect to monomial orders. As an applica- tion we recover the existence of the generic fan introduced by Romer and Schmitz for studying generic tropical varieties. We also consider general initial ideals with respect to weights and show, in analogy to the fact that generic initial ideals are Borel-fixed, that these are fixed under the action of certain Borel subgroups of the general linear group.

The argument cycle and the coamoeba

We investigate the coamoeba of a complex algebraic variety V ⊂ (ℂ*) n through the study of initial forms of the defining ideal. By use of a universal Gröbner basis, we prove that the closure of the coamoeba is included in the union of coamoebas corresponding to all initial ideals. We also study complete intersections V of dimension n/2 more closely to get a lower bound for the multiplicity in V of a given point θ on the n:th torus. For this purpose, we associate a certain algebraic cycle, the argument cycle, to V and θ , and study its homology. In particular, we give a method to approximate the coamoeba when n = 2.

Gorenstein algebras presented by quadrics

We establish restrictions on the Hilbert function of standard graded Gorenstein algebras with only quadratic relations. Furthermore, we pose some intriguing conjectures and provide evidence for them by proving them in some cases using a number of different techniques, including liaison theory and generic initial ideals.

Generic initial ideals of singular curves in graded lexicographic order

In this paper, we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order. Let C be a singular irreducible projective curve of degree d⩾5 with the arithmetic genus ρa(C) in Pr where r⩾3. If M(IC) is the regularity of the lexicographic generic initial ideal of IC in a polynomial ring k[x0,…,xr] then we prove that M(IC) is 1+(d−12)−ρa(C) which is obtained from the monomialxr−3xr−1(d−12)−ρa(C), provided that dimTanp(C)=2 for every singular point p∈C. This number is equal to one plus the number of secant lines through the center of general projection into P2. Our result generalizes the work of J. Ahn (2008) [1] for smooth projective curves and that of A. Conca and J. Sidman (2005) [9] for smooth complete intersection curves in P3. The case of singular curves was motivated by A. Conca and J. Sidman (2005) [9, Example 4.3]. We also provide some illuminating examples of our results via calculations done with Macaulay 2 and Singular (Decker et al., 2011 [10], Grayson and Stillman [16]).

Finite lattices and Gröbner bases

AbstractGröbner bases of binomial ideals arising from finite lattices will be studied. In terms of Gröbner bases and initial ideals, a characterization of finite distributive lattices as well as planar distributive lattices will be given.

Minimal graded resolutions of reverse lexsegment ideals

Let k be a field, and let S = k[x1, …, xn] be the polynomial ring in x1, …, xn with coefficients in the field k. We study the minimal graded free S-resolutions of reverse lexsegment ideals of S. We discuss the extremal Betti numbers of initial reverse lexsegment ideals of S. Moreover, we analyze all reverse lexsegment ideals with linear resolution.

INDEPENDENT SETS FROM AN ALGEBRAIC PERSPECTIVE

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.

Scarf lattice ideals

This paper deals with the Scarf property of lattice ideals initiated by Peeva and Sturmfels[10], [11]. We will present a Scarf lattice ideal that is neither generic nor of codimension 2 and show that this property gives rise to several algebraic and combinatorial properties. In particular, we prove that for monomial curves, this property coincides with the notion of genericity, and that certain Scarf lattice ideals can have certain Scarf initial ideals.

Depth of edge rings arising from finite graphs

Let G G be a finite graph and K [ G ] K[G] the edge ring of G G . Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers f f and d d with 7 ≤ f ≤ d 7 \leq f \leq d , there exists a finite graph G G on [ d ] = { 1 , … , d } [d] = \{ 1, \ldots , d \} with depth K [ G ] = f \textrm {depth}\, K[G] = f and with Krull-dim K [ G ] = d \textrm {Krull-dim}\, K[G] = d .

In Search of a Wide-Angle Lens

In Search of a Wide-Angle Lens Harold Braswell (bio) What issues should bioethics be looking at in the next forty years? Rather than take on new issues, I believe bioethicists should rethink our approach to bioethical topics more generally. Doing so will require refashioning the field itself, but such a reinvention is the only way we can help bioethics live up to its initial ideals and be relevant to our society. Is Bioethics Necessary? Thinking about the future of bioethics should begin with a fundamental question: Is bioethics even necessary? Most bioethicists would certainly think so, and they might find confirmation in histories of the field like David Rothman's Strangers at the Bedside, in which bioethics was a response to the collapse of doctor-patient relations in the midtwentieth century United States.1 Here, the field emerged to protect vulnerable populations from exploitative doctors and scientists. As such, it was allied with the civil rights and feminist movements. What could be more necessary than that? But this rosy view of the field's development has been highly questioned. Roger Cooter has argued that bioethics appeared to allow doctors to maintain control over the provision of health care; rather than a progressive movement aligned with civil rights and feminism, it was a reaction to [End Page 19] the anxiety these movements generated in physicians.2 His reading would seem to be substantiated by Susan Lederer's account of the generation of the Helsinki Declaration, in which the opposition of U.S. doctors to the declaration's prohibitions on experimentation on children and prisoners was connected to the pharmaceutical industry's desire to maximize profit.3 More recently, various authors have illustrated how foundational bioethical concepts like "autonomy" and "informed consent" are used in ways that exploit vulnerable populations, such as the poor and disabled.4 These sources show that we bioethicists, rather than looking for new issues, would do better to look at ourselves. The problem is the way we have framed our approaches to the field's key topics. We have been obsessed with questions of abortion, euthanasia, stem cell research, and the like. But the very framing of these issues in bioethical discourse can obscure the underlying forces that create the problems to begin with, whether these forces be the economic organization of society, internalized discrimination against the disabled, the epistemology of medicine, or (and most likely) some combination of all three. We need to find ways to understand the larger problems that create the "old" issues—problems in which we can productively intervene. A New Model for Bioethics Education Here we encounter a significant institutional problem: U.S. bioethics education remains tethered to a narrowly defined training regimen that is inadequate to explore the underlying causes of the field's problems. As such, it frequently does not prepare students to understand the issues they are supposed to resolve. Consequently, our primary concern should be reformulating bioethics education in a way that gives students the tools to understand and investigate the field's concerns. Bioethics institutes, rather than being oriented around the transmission of rule-based knowledge, should provide a robust, interdisciplinary curriculum that produces thinkers capable of exploring the complex context of bioethical issues and proposing mechanisms to resolve present conflicts. The goal of this curriculum would not be to prepare thinkers to "take sides" on current bioethical debates, but instead to explore how the framing of these debates can mystify the real issues at play. Exploring the dynamics of problem constitution would be the first step toward problem solving.5 This would make it possible to productively intervene in debates whose terms have become stagnant. Students emerging from such programs would be prepared to comment on how the dichotomy between the secular "pro-choice" and the religious "pro-life" perspectives that frames our discussions of abortion and euthanasia obscures how the economic ideologies of both sides in the debate might encourage people to abort and engage in euthanasia, even as they negate whatever "freedom" might exist in that choice. The issue would be discussed in more than just economic terms: One could talk about how cultural discrimination against the disabled can be reflected both in...

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals I t of the extended Hankel matrices. Denote by c -chain a sequence a 1 , … , a k with a i + c < a i + 1 for all i = 1 , … , k − 1 . Using the results of c -chain, we solve the membership problem for the symbolic powers I t ( s ) and we compute the primary decomposition of the product I t 1 ⋯ I t k of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product I t 1 ⋯ I t k has a linear resolution and the multi-homogeneous Rees algebra Rees ( I t 1 , … , I t k ) is defined by a Gröbner basis of quadrics.

Gröbner bases of ideals cogenerated by Pfaffians

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen–Macaulay. We also provide a formula for computing their multiplicity.

The Ratliff-Rush Closure of Initial Ideals of Certain Prime Ideals

We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.

Alternatives for testing total dual integrality

In this paper we provide characterizing properties of totally dual integral (TDI) systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system’s dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick’s sufficient condition for the TDI property and Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.

Them Who Kill the Body: Christian Ideals and Political Realities in the Interior of Southern Africa during the 1850s

This article considers the changing political significance of Christianity in the interior of southern Africa during the 1850s, focusing primarily on the views of Tswana rulers, converts and others within their communities, and secondarily on attempts by European missionaries to reconcile their service both to African communities and to European expansion, which compelled them to articulate a rationale for their civilising mission. The article historicises the process whereby Christianity lost its initial universalistic ideals and became politicised by African-European competition, with divine sanction being claimed by one side or another. That process was accompanied by considerable debate and doubt: the separation of believers from unbelievers, faith from works and souls from bodies was far from clear or certain. Although African converts would ultimately continue to pursue the promise of equality offered by Christian teachings, they would do so more as individuals than as intact African communities, surrendering their bodies to European rule while entrusting their souls to the care of God.

Generic initial ideals and fibre products

We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute explicitly the generic initial ideal of a fibre product in a special case. We also prove that the fibre product of two graded ideals is componentwise linear if and only if both ideals have this property.

Generic initial ideals of some monomial complete intersections in four variables

Let R = K[x 1, x 2, x 3, x 4] be the polynomial ring over a field of characteristic zero. For the ideal $${(x_1^a, x_2^b, x_3^c, x_4^d) \subset R}$$ , where at least one of a, b, c and d is equal to two, we prove that its generic initial ideal with respect to the reverse lexicographic order is the almost revlex ideal corresponding to the same Hilbert function.

Generic Initial Ideals, Graded Betti Numbers, and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are: 1. Let I be an ideal of R = K[x 1, x 2,…, x n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.

Quadratic Gröbner bases for smooth 3×3 transportation polytopes

The toric ideals of 3×3 transportation polytopes $\mathsf{T}_{\mathbf{rc}}$ are quadratically generated. The only exception is the Birkhoff polytope B 3. If $\mathsf{T}_{\mathbf{rc}}$ is not a multiple of B 3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3×3 transportation polytopes.