Border Basis Research Articles

Re-embeddings of affine algebras via Gröbner fans of linear ideals

Given an affine algebra R=K[x1,⋯,xn]/I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=K[x_1,\\dots ,x_n]/I$$\\end{document} over a field K, where I is an ideal in the polynomial ring P=K[x1,⋯,xn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P=K[x_1,\\dots ,x_n]$$\\end{document}, we examine the task of effectively calculating re-embeddings of I, i.e., of presentations R=P′/I′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=P'/I'$$\\end{document} such that P′=K[y1,⋯,ym]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'=K[y_1,\\dots ,y_m]$$\\end{document} has fewer indeterminates. For cases when the number of indeterminates n is large and Gröbner basis computations are infeasible, we have introduced the method of Z-separating re-embeddings in Kreuzer et al. (J Algebra Appl 21, 2022) and Kreuzer, et al. (São Paulo J Math Sci, 2022). This method tries to detect polynomials of a special shape in I which allow us to eliminate the indeterminates in the tuple Z by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples Z can be found using the Gröbner fan of the linear part of I. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the case of binomial linear ideals using a cotangent equivalence relation. Finally, we apply the improved technique in the case of the defining ideals of border basis schemes.

Algorithms for checking zero-dimensional complete intersections

Given a 0-dimensional affine K-algebra R=K[x1,…,xn]∕I, where I is an ideal in a polynomial ring K[x1,…,xn] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

Исторический опыт формирования российско-финляндской границы: теоретические аспекты становления (и формирования) международно-правового регулирования

The problems of the historical aspects of the formation of interstate borders, which continue to be the focus of attention of representatives of the scientific and practical community, are analyzed. The above is due, among other things, to the specifics of the geopolitical position of individual border territories, which were the subject of disagreements and disputes. From the standpoint of the theory and methodology of law, the problems around state borders are dictated by the growing objective and natural needs of states and the legally formalized directions of the latter's policy in this area. Attempts have been made to consider some historical aspects of the formation of the legal basis of the Russian-Finnish border. Against the backdrop of increased rivalry between key centers of power, growing contradictions and dynamic changes in international relations, a new rethinking of the historical process of legal registration of the establishment of the border between the two countries will become of particular relevance. An analysis of the early historical patterns and the genesis of the formation of the interstate border makes it possible to build it in line with the revision and reassessment of certain aspects of the logic of the formation of the border space within the framework of the historical and legal process. It is concluded that in modern conditions, on the one hand, the readiness to develop in-depth cooperation within the framework of the emerging new model of behavior in this area and, on the other hand, the moods traditionally taking place in certain political and public circles of Finland regarding the return of the lost Karelian lands determine the use of the most stable border regimes and adequate political and legal mechanisms, procedures to avoid the negative impact of the factor of increasing the process of militarization of the border and individual risks. In the context of a possible change in the policy of the two countries, the theory of legal regulation of the issues under study is of great importance, without which it seems impossible to achieve a balance of the multifaceted interests of states.

Taxation of Insurance Premiums in the EU Cross-Border Insurance Business

As in many other cases of business activity, the performance of insurance activity involves various types of tax charges. Insurance contracts, having been excluded from the scope of value-added tax at the EU level, are subjected to taxes specific to insurance activity, namely insurance premium taxes, which are discussed in the article. However, insurance premium tax regulations are not harmonised at the EU level and vary immensely from one country to another. In fact, in some EU countries, including Poland, no such taxes have been implemented whatsoever. The issue discussed in the article is related to the fact that an increasing number of insurance companies perform insurance activity on a cross border basis. From this point of view, although there is no such tax in Poland, it may be of great importance to Polish insurance companies that insure risks located outside Poland. The regulations concerning the insurance premium tax depend on internal decisions of individual European Union countries, both in terms of the introduction of such tax, exemptions, and rates, and they may differ significantly across those jurisdictions. Due to the nature of cross-border activity and the rules on the localisation of the risk associated with the taxation of premiums, it was not clear whether and to what extent, in the case of insurance of risks located outside Poland, a tax obligation arises in respect of insurance premiums. These questions were addressed by the Court of Justice of the EU in several cases (C-118/96, C191/99, C-243-11). The CJEU decisions have revealed the multifaceted complexity inherent to the taxation of cross border insurance contracts.

Differentiation of adenocarcinoma in situ with alveolar collapse from minimally invasive adenocarcinoma or invasive adenocarcinoma appearing as part-solid ground-glass nodules (≤ 2cm) using computed tomography.

To investigate the differentiating computed tomographic (CT) features between adenocarcinoma in situ (AIS) with alveolar collapse and minimally invasive adenocarcinoma (MIA) or invasive adenocarcinoma (IA) appearing as part-solid nodules. A total of 147 consecutive patients with 157 pathology-confirmed part-solid ground-glass nodules (GGNs) ≤ 20mm without other pathological condition such as inflammation and fibrosis who underwent chest CT were included. The 157 part-solid GGNs included 33 (21.02%) pathologically confirmed AISs with alveolar collapse. Multivariate analysis revealed that smaller lesion size (odds ratio [OR] 0.671), and well-defined border (OR 5.544), concentrated distribution (OR 7.994), and homogeneity of the solid portion (OR 4.365) were significant independent predictors for differentiating AIS with alveolar collapse from MIA (P < 0.05) with excellent accuracy (area under receiver operating characteristic [ROC] curve, 0.902). Multivariate analysis revealed that smaller lesion size (OR 0.782), and size (OR 0.821), well-defined border (OR 5.752), and homogeneity of solid portion (OR 6.182) were significant independent predictors differentiating AIS with alveolar collapse from IA (P < 0.05) with excellent accuracy (area under ROC curve 0.910). Among part-solid GGNs, AIS with alveolar collapse can be accurately differentiated from MIA on the basis of smaller lesion size, well-defined border, concentrated distribution, and homogeneity of solid portion, and from IA according to smaller lesion size, and smaller size, well-defined border, and homogeneity of solid portion.

Polynomial-division-based algorithms for computing linear recurrence relations

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence.Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations.A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial.As an addition from the paper published at the ISSAC conference, we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gröbner basis.All these algorithms have been implemented in Maple and we report on our comparisons.

Border Basis of an Ideal of Points and its Application in Experimental Design and Regression

Border Basis of an Ideal of Points and its Application in Experimental Design and Regression

Computing subschemes of the border basis scheme

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Borders as opposed to so-called geography: which should be used to classify isolated ventricular septal defects?

Ventricular septal defects can be classified according to their borders or according to the fashion in which they open to the right ventricle, so-called geography. As yet, there is no consensus as to how they should be classified. In an attempt to achieve agreement, the International Society for Nomenclature of Congenital and Paediatric Heart Disease, in 2018, proposed a system incorporating both approaches. We have assessed the subjectivity of their suggested terms hoping to determine their suitability in the desired universal system for classification. We examined 212 specimens held in the archive of Birmingham Women's and Children's Hospital. Each defect was described by 3 independent examiners on the basis of borders and their relationship to the landmarks of the right ventricle. The interobserver agreement was then calculated using Fleiss' method. Calculations to assess interobserver agreement showed that the examiners were more likely to agree on the borders of the defects than their so-called geography (κ = 0.804 vs κ = 0.518). The landmarks of the right ventricle proved to be highly variable such that the application of 'geographic' terms to hearts with perimembranous defects proved particularly challenging. Interobserver agreement is lower when using terms based on 'geography' as opposed to borders. Whilst providing important morphological detail, the terms based on right ventricular landmarks are highly subjective. They should not be prioritized in a universal system of classification. Instead, the defects can be classified simply by using 'perimembranous', 'muscular', or 'doubly committed and juxta-arterial' as first-order terms.

A Signature Based Border Basis Algorithm

The Border Basis Algorithm (BBA) still suffers from the lack of analogues of Buchberger’s criteria for avoiding unnecessary reductions. In this paper we develop a signature based technique which provides a first remedial step: signature bounds allow us to recognize multiple reductions of the same ancestor polynomial. The new signature based algorithm is then combined with the Boolean BBA for ideals of Boolean polynomials. Experiments show that it is at least 5 times faster than the standard Boolean BBA.

A Contraction Mapping Method in Digital Image Processing

The topological features of the objects or digital image pictures are characterized by digital topology. A digital image picture is a distinctive arrangement of numbers which are non negative. Decomposing an image picture into its constituent components and investigating its several characteristics with fundamental elements is generally coined as digital image processing. In investigating the fundamental constituents of images, the separation of connected segments are established to enquire the adjacency relationship. During this course of tracking, thinning and coding them, it is kept in mind that the specification of connectedness of the pictures remains unaltered. The characteristics of the constituent subsets and relationships may be stipulated when the image is fragmented into its elementary constituents. Some features of the subsets of these constituent parts are based on their respective positions. Thus, the basic idea for image processing is the primary topological characteristics of digital images like adjacency, connectedness, etc. The fixed point theorems associated to certain kinds of contraction mappings can be utilised in the field of engineering science and technology as computational technique to provide an exclusive programme to explore various problems. Parallel and distributed computation, modeling, simulation and digital image processing are few notables among these techniques. In digital image processing digital contraction mapping is defined and then existence of the solution and its uniqueness is obtained using the fixed point theorem concerned, which is the mathematical basis of border following, thinning and contour filling of a digital image picture. In digital image processing the applicability of fixed point theorem and contraction mappings as a computational technique has been grazed well. To further broaden its pertinency in image processing our interest is to delve into some of contraction mappings as a significant mechanism.

Computing Coupled Border Bases

Kehrein and Kreuzer (J Pure Appl Algebra 205(2):279–295, 2006) presented an algorithm, referred to as the BorderBasis algorithm, for computing border bases. The main objective of this paper is to improve this computation by applying new structures from the theory of Grobner bases. In doing so, based on the method developed by Gao et al. (Math Comput 85(297):449–465, 2016), a new variant of the BorderBasis algorithm is proposed to compute simultaneously a border basis and also a Grobner basis for the syzygy module of the input polynomials (this pair of bases will be called a coupled border basis). The new algorithm, and also the original form of the BorderBasis algorithm, have been implemented in the computer algebra systems Maple and ApCoCoA. We compare the performance of our implementation of these algorithms via a set of benchmark polynomials. Our experiments show that our algorithm performs more efficiently than the original BorderBasis algorithm.

On the regularity of the monomial point of a border basis scheme

The border basis scheme $${\mathbb {B}}_{{\tiny {\mathcal {O}}}}$$ parametrizes all 0-dimensional ideals in $$K[x_1,\ldots ,x_n]$$, where K is an arbitrary field, which have a border basis with respect to a given order ideal of terms $${\mathcal {O}}$$. Its vanishing ideal $$I({\mathbb {B}}_{{\tiny {\mathcal {O}}}})$$ is generated by quadratic equations which are easy to describe, and it contains a unique monomial point, namely the point corresponding to the monomial ideal generated by the terms outside $${\mathcal {O}}$$. Based on a detailed study of the generators of $$I({\mathbb {B}}_{{\tiny {\mathcal {O}}}})$$, we describe a K-basis of the cotangent space $${\mathfrak {m}}/{\mathfrak {m}}^2$$, where $${\mathfrak {m}}$$ is the maximal ideal of the monomial point. Moreover, we provide an efficient algorithm to compute such a basis and use it to characterize the regularity of the monomial point.

Game Theory in Israel’s Relation with the Palestinian Authority: Study of Israel’s Withdrawal to Borders of the Year 1967, a Model (1993-2018)

This study examines the concept of game theory, its objectives and its elements. It also reviews the Israeli Palestinian treaties and the international resolutions related to the issue of the Israeli withdrawal from the Palestinian territories on the basis of borders of 4 June 1967 including Jerusalem, as one of the core issues for reaching a final status agreement between the Palestinian and Israeli sides, the declaration of the establishment of a Palestinian state, ending the state of conflict between the two sides and reaching the stage of peaceful coexistence. The review involves clarifying the instruments and the techniques upon which Israel propped in applying this theory in its relation with the Palestinian Authority, and analysing the extent of gains or less achieved by both sides in this issue, in accordance with the concept of zero theory, or non- zero theory based games. The study had deduced that Israel confined in its strategy to employing the method of zero game with the Palestinian National Authority considering itself the stronger party who is able to exerting its pressures to achieve the greatest amount of gains, but the Palestinian leadership could not, till this moment achieve any gains rather the relinquishment it made by signing the Oslo Agreement were followed by subsequent relinquishment.

Computing all border bases for ideals of points

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

Truncated normal forms for solving polynomial systems

In this poster we present the results of [10]. We consider the problem of finding the common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the cokernel of a resultant map. This leads to what we call Truncated Normal Forms (TNFs). Algorithms for generic dense and sparse systems follow from the classical resultant constructions. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm. The numerical experiments show that the methods allow to compute all zeros of challenging systems (high degree, with a large number of solutions) in small dimensions with high accuracy.

Refugees in Greece: the Greeks as ‘refugees’

The state of economic emergency under which Greece has been put for the past eight years throws into relief the basic antinomy inherent in democracy. This pertains to the exercise of national sovereignty on the basis of borders whose safeguarding, however, implements a network of state practices of control and selection of populations both ‘within’ and ‘outside’. In Greece, under the memoranda imposed by the Tetroika (involving four institutions: IMF, ECB, ESM and EC, not three as in Troika), extreme austerity has created ‘superfluous’ populations within their own country. The internal shifting of the borders that produces exclusion, sustained by parliamentary dictates and intense supervision because of the Tetroika's policies at the national level, goes along with the stiffening of external border control by the state in a strategy of deterrence against the entry of war refugees into the country. At the same time, the regulation of ‘superfluous’ refugee populations replicates the biopolitical model of EU governance introduced to the national ‘body’. Therefore, the Greek radical Left has to demonstrate that both dispossessed Greek subjects and refugees are victims of globalized capitalism, distance itself from humanitarianism and politicize solidarity by creating the terms for a common struggle.

Solving Polynomial Systems via Truncated Normal Forms

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the null space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous and multi-homogeneous cases are treated. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm and show numerical results.

The Genealogy of Culturalist International Relations in Japan and Its Implications for Post-Western Discourse

This paper aims to introduce a neglected methodology from Japanese international relations (IR) – the culturalist methodology – to Anglophone specialists in IR. This methodology is neglected not only by an Anglophone audience but also by Japanese IR scholars. I argue here that despite this negligence, the culturalist methodology has great potential to contribute to contemporary post-Western international relations theory (IRT) literature by posing radical questions about the ontology of IR, as it questions not only the ontology of Western IR, but also the IR discourses developed in the rest of the world. Consequently, in understanding and imagining the contemporary world, I clarify the importance of perceptions based on what, in Japan, are commonly called ‘international cultural relations’ (kokusai bunka) and ‘regional history’ (chiikishi). I also indicate how our perceptions of the world are limited by the Westphalian principles of state sovereignty and non-intervention among ‘equal’ nations on the basis of state borders. While historical understanding is widely recognised as an important approach to contemporary IR, its scope is limited by its universalised principles.

Local Bézout theorem for Henselian rings

In this paper we prove what we call Local Bezout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and m-adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature.