Kei-ichiro Iima, Yuji Yoshino
We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Groebner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals.
We introduce the theory of monoidal Groebner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Groebner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness theorems in commutative algebra and its applications. A major result of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. We use this machinery to give new proofs of some classical finiteness theorems in algebra…
We introduce the notion of monomial group action and study some of its consequences for Groebner basis theory. As an application we prove a conjecture of V. Batyrev and O. Popov describing the Cox rings of Del Pezzo surfaces (of degree at least 3) as quotients of a polynomial ring by an ideal generated by quadrics.
In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and regularity of A and R.
We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both …
Let $I\subseteq S=K[x_1,\ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $\operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/\operatorname{in}_<(I)$ have the same connectedness dimension. We also show that graphs related to connectedness of these quotient rings have the same number of components. We also provide consequences regarding Lyubeznik numbers. We obtain these results by furthering the study of connectedness modulo a parameter in a local ring.
We give an introduction to the theory of determinantal ideals and rings, their Groebner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras.
Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on rational function fields of two variables that are suitable for this framework. For these valuations, we explicitly compute the image of an underlying polynomial ring and use this to perform computations concerning ideals in the polynomial ring. Interestingly, for these valuations, some ideals have a fini…
Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Groebner bases, but we also give examples showing that the resulting Groebner bases can be substantially smaller than traditional Groebner bases. In the case K = Q with the p-adic …
We introduce a general constructive method to find a p-basis (and the Ulm invariants) of a finite Abelian p-group M. This algorithm is based on Groebner bases theory. We apply this method to determine the additive structure of indecomposable modules over the following Dedeking-like rings: ZCp, where Cp is the cyclic group of order a prime p, and the p-pullback Z --> Zp <-- Z of Z+Z.
Small Groebner Fans of Ideals of Points
本文旨在识别多项式环中具有唯一约化Groebner基的点理想类并提供构造方法，同时探讨具有相同Groebner扇基数的理想对。
In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Groebner bases based on the articles by Arnold (cf. [A03]), Idrees, Pfister, Steidel (cf. [IPS11]) and Noro, Yokoyama (cf. [NY12], [Y12]). In fact, the first algorithm that mainly uses the given symmetry, improves the necessary…
The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${\bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariski's factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.
This short note is the generalization of Faugere F4-algorithm for polynomial rings with coefficients in Euclidean rings. This algorithm computes successively a Groebner basis replacing the reduction of one single s-polynomial in Buchberger's algorithm by the simultaneous reduction of several polynomials.
It is an open question whether tight closure commutes with localization in quotients of a polynomial ring in finitely many variables over a field. Katzman showed that tight closure of ideals in these rings commutes with localization at one element if for all ideals I and J in a polynomial ring there is a linear upper bound in q on the degree in the least variable of reduced Groebner bases in reverse lexicographic ordering of the ideals of the form J + I^{[q]}. Katzman conjectured that this property would always be satisfied. In this paper we prove several cases of Katzman's conjecture. We also…
We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The ob…
Let I be a homogeneous ideal of a polynomial ring S. We prove that if the initial ideal J of I, w.r.t. a term order on S, is square-free, then the extremal Betti numbers of S/I and of S/J coincide. In particular, depth(S/I)=depth(S/J) and reg(S/I)=reg(S/J).
On modular computation of Groebner bases with integer coefficients
针对整数系数多项式环中递增理想并集的 Gröbner 基计算问题，论文提出利用有理化理想及模 $m$ 理想的已知 Gröbner 基进行模计算的算法。
In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Groebner bases via Buchberger's Algorithm.
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Groebner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Groebner basis determined by the cycles. It turn…
Let $I$ be an ideal of the polynomial ring $A[x]=A[x_1,...,x_n]$ over the commutative, noetherian ring $A$. Geometrically $I$ defines a family of affine schemes over $\Spec(A)$: For $\p\in\Spec(A)$, the fibre over $\p$ is the closed subscheme of affine space over the residue field $k(\p)$, which is determined by the extension of $I$ under the canonical map $σ_\p:A[x]\to k(\p)[x]$. If $I$ is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is $\sigI$. For a chosen term order this ideal has a unique reduced Gröbner basis which is known to contain consi…
The aim of this note is to present an easy proof of Hilbert's Nullstellensatz using Groebner basis. I believe, that the proof has some methodical advantage in a course on Groebner bases. Key words: Hilbert's Nullstellensatz, Groebner bases.
In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.
This paper presents an algorithm for computing Groebner bases based upon labeled polynomials and ideas from the algorithm F5. The main highlights of this algorithm compared with analogues are simplicity both of the algorithm and of the its correctness proof achieved without loss of the efficiency. This leads to simple implementation which performance is in par with more complex analogues
Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Groebner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications. In this paper we compare Groebner basis schemes with border basis schemes. It is…
Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Groebner basis of the ideal ( a, b, p^e), is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to t…
In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.
The aim of this note is to discuss the following quite queer Problem: \noindent GIVEN \noindent i) the free non-commutative polynomial ring, ${\Cal P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (public)}, \noindent ii) a bilateral ideal ${\sf I}\subset {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (private)}, \noindent iii) a finite set $G := \{g_1,\ldots,g_l\}\subset{\sf I}$ of elements of the ideal ${\sf I}$ {\em (public)}, \noindent a noetherian semigroup term-ordering $\prec,$ {\rm (private)}, on the word semigroup ${\Cal T} := < X_1,\ldots,X_n>$, \noindent COMPUTE \noindent --a finite s…
We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Moell…
Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive…
Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gröbner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.
In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.
Faugere's F5 algorithm computes a Groebner basis incrementally, by computing a sequence of (non-reduced) Groebner bases. The authors describe a variant of F5, called F5C, that replaces each intermediate Groebner basis with its reduced Groebner basis. As a result, F5C considers fewer polynomials and performs substantially fewer polynomial reductions, so that it terminates more quickly. We also provide a generalization of Faugere's characterization theorem for Groebner bases.
The theory of Groebner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Groebner Basis, and indeed in finding alternative methods of computing Groebner Bases. Two of these methods include the Groebner Walk method and the computation of Involutive Bases. By the mid 1980's, Buchberger's work had been generalised for noncommutative polynomial rings by Bergman and Mora. This thesis provides the corresponding generalisati…
We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Groebner bases for these ideals. This allows us to unify and generalize some results in algebraic statistics.
We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a commutative field K of characteristic zero. In order to reach explicit representatives of some generators of the quotient space Z^k/B^k of cocycles Z^k modulo coboundaries B^k, we apply Groebner bases techniques (in the appropriate linear setting) and take advantage of their strength. Moreover, when the considered Lie (super) algebras enjoy a grading -- a case which often happens both in representation theory and in differential geometry --, all cohomology spaces Z^k…
A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Grobner bases have recently been employed as a selection tool for polynomi…
The purpose of this paper is twofold. In the first part we concentrate on hyperplane sections of algebraic schemes, and present results for determining when Gröbner bases pass to the quotient and when they can be lifted. The main difficulty to overcome is the fact that we deal with non-homogeneous ideals. As a by-product we hint at a promising technique for computing implicitization efficiently. In the second part of the paper we deal with families of algebraic schemes and the Hough transforms, in particular we compute their dimension, and show that in some interesting cases it is zero. Then w…
We give an introduction to the theory of initial ideals and initial algebras with emphasis on the transfer of structural properties.
In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.
This article is partly a survey and partly a research paper. It tackles the use of Groebner bases for addressing problems of numerical semigroups, which is a topic that has been around for some years, but it does it in a systematic way which enables us to prove some results and a hopefully interesting characterization of the elements of a semigroup in terms of Groebner bases.
We show how tropical varieties of ideals I over a field K with non-trivial valuation can be traced back to tropical varieties of ideals in R[[t]][x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that generators are polynomial in t. While doing so, we also comment on the computation of the Groebner polytope structure and p-adic Groebner bases using our framework. All algorithms solely rely on existing standard basis techniques.
In this paper we use Groebner bases theory in order to determine planarity of intersections of two algebraic surfaces in ${\bf R}^3$. We specially considered plane sections of certain type of conoid which has a cubic egg curve as one of the directrices. The paper investigates a possibility of conic plane sections of this type of conoid.
This paper presents algorithms for computing the Groebner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Groebner bases of the ideal. Our algorithms are based on a uniform definition of the Groebner fan that applies to both homogeneous and non-homogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Groebner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memory-less reverse search procedure. The significance of this follows from the fact that Groebner fans ar…
Groebner.jl: A package for Gröbner bases computations in Julia
本文提出Julia包Groebner.jl，基于F4算法计算Gröbner基，支持模素数整数与有理数运算及多线程，结合符号计算技术与Julia类型系统，在性能与可扩展性上达到或超越现有最先进水平。
Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner basis of the ideal of elementary symmetric polynomials in $n$ variables of degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further generalize their findings by using symbolic computation and experimentation to construct the reduced Groebner basis for the ideal generated by the elemen…
Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable sum problems and shows that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Groebner bases.
We present a survey on the developments related to Groebner bases, and show explicit examples in CoCoA. The CoCoA project dates back to 1987: its aim was to create a "mathematician"-friendly computational laboratory for studying Commutative Algebra, most especially Groebner bases. Always maintaining this "friendly" tradition, the project has grown and evolved, and the software has been completely rewritten. CoCoA offers Groebner bases for all levels of interest: from the basic, explicit call in the interactive system CoCoA-5, to problem-specific optimized implementations, to the computer--comp…
We outline a generalization of the Groebner fan of a homogeneous ideal with maximal cells parametrizing truncated Groebner bases. This "truncated" Groebner fan is usually much smaller than the full Groebner fan and offers the natural framework for conversion between truncated Groebner bases. The generic Groebner walk generalizes naturally to this setting by using the Buchberger algorithm with truncation on facets. We specialize to the setting of lattice ideals. Here facets along the generic walk are given by unique (facet) binomials. This along with the representation of binomials as integer v…
In this paper we describe an efficient involutive algorithm for constructing Groebner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Groebner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be output without any extra computational costs. We also discuss some accounts of experimental superiority of the involutive algorithm over Buchberger's algorithm.
This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n. A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes. We also discuss regular tilings by ribbon L n-ominoes, n odd, for rectangles and more general regions. We show that in this case obstructions appear that are not detected by signed tilings.
Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the first question. It is well-known that G is a Groebner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m-1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results.…
A topological space TO(S) of total orderings on any given set S is introduced and it is shown that TO(S) is compact if S is countable. The set NO(N) of all normal orderings of the nth Weyl algebra W is a closed subspace of TO(N), where N is the set of all normal monomials of W. Hence NO(N) is compact and, as a consequence of this fact and by a division theorem valid in W, we give a proof that each left ideal of W admits a universal Groebner basis.
For a given monomial ideal $J \subset k[x_1, \ldots, x_n]$ and a given monomial order $\prec$, the moduli functor of all reduced Gröbner bases with respect to $\prec$ whose initial ideal is $J$ is determined. In some cases, such a functor is representable by an affine scheme of finite type over $k$, and a locally closed subfunctor of a Hilbert scheme. The moduli space is called the Gröbner basis scheme, the Gröbner strata and so on if it exists. This paper introduces an alternative procedure for explicitly constructing a defining ideal of the Gröbner basis scheme and its Zariski tangent spaces…
We give upper bounds for the solving degree and the last fall degree of the polynomial system associated to the HFE (Hidden Field Equations) cryptosystem. Our bounds improve the known bounds for this type of systems. We also present new results on the connection between the solving degree and the last fall degree and prove that, in some cases, the solving degree is independent of coordinate changes.
The toric ideals of $3\times 3$ transportation polytopes $T$ are quadratically generated. The only exception is the Birkhoff polytope $B_3$. If $T$ is not a multiple of $B_3$, these ideals even have squarefree quadratic initial ideals. This class contains all smooth $3 \times 3$ transportation polytopes.
A set of polynomials G in a polynomial ring S over a field is said to be a universal Groebner basis, if G is a Groebner basis with respect to every term order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that the set of the maximal minors of a matrix X of variables is a universal Groebner basis. Boocher recently proved that any initial ideal of the ideal of maximal minors of X has a linear resolution. In this paper we give a quick proof of the results mentioned above. Our proof is based on a specialization argument. Then we show that similar statements hold in a more g…
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. By associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。