Groebner bases, initial ideals and initial algebras

Universal Groebner bases and Cartwright-Sturmfels ideals

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.

点理想的Small Groebner扇

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。

Groebner Bases for Transportation Polytopes

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.

Groebner bases of symmetric ideals

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…

Indispensable Hibi relations and Groebner Bases

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.

A Combinatorial Approach to the Groebner Bases for Ideals Generated by Elementary Symmetric Functions

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…

Groebner bases, monomial group actions, and the Cox rings of Del Pezzo surfaces

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.

Contracted ideals and the Groebner fan of the rational normal curve

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.

Partition identities and application to infinite dimensional Groebner basis and viceversa

In the first part of this article, we consider a Groebner basis of the differential ideal {x_1^2} with respect to "the" weighted lexicographical monomial order and show that its computation is related with an identity involving the partitions that appear in the first Rogers-Ramanujan identity. We then prove that a Grobener basis of this ideal is not differentially finite in contrary with the case of "the" weighted reverse lexicographical order. In the second part, we give a simple and direct proof of a theorem of Nguyen Duc Tam about the Groaner basis of the differential ideal {x_1y_1}; we the…

Square-free Groebner degenerations

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).

Efficiently computing Groebner bases of ideals of points

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…

Involutive Algorithms for Computing Groebner Bases

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.

Two way subtable sum problems and quadratic Groebner bases

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.

Computing Groebner Fans

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 bases for families of affine or projective schemes

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…

A non-regular Groebner fan

The Groebner fan of an ideal $I\subset k[x_1,...,x_n]$, defined by Mora and Robbiano, is a complex of polyhedral cones in $R^n$. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of $I$ as the term order varies. If $I$ is homogeneous the Groebner fan is complete and is the normal fan of the state polytope of $I$. In general the Groebner fan is not complete and therefore not the normal fan of a polytope. We may ask if the restricted Groebner fan, a subdivision of $R_{>=0}^n$, is regular i.e. the normal fan of a polyhedron. The main result of this paper is a…

计算理想插值的Gröbner基

Computing Groebner bases of ideal interpolation

本文提出一种通过泰勒展开与高斯消元计算理想插值中消失理想约化Gröbner基的多项式时间算法，并在单点及部分多点插值中优于MMM算法。

Groebner bases of nested configurations

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.

An extension of Buchberger's criteria for Groebner basis decision

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.…

Computing tropical varieties over fields with valuation

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.

Finite Groebner bases in infinite dimensional polynomial rings and applications

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…

Groebner bases and regularity of Rees algebras

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.

Computing Groebner bases of ideals of few points in high dimensions

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…

Groebner bases for the polynomial ring with infinite variables and their applications

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.

Connectedness of square-free Groebner Deformations

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.

Value Monoids of Zero-Dimensional Valuations of Rank One

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…

Construction of the moduli space of reduced Groebner bases

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…

Groebner bases over fields with valuations

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 …

KRS and determinantal ideals

The first sections contain a survey of the application of the Knuth-Robinson-Schensted corerspondence to the computation of Groebner bases of determinantal ideals. We also set up a conceptual framework for this application in terms of so-called "KRS invariants". Then we show that the initial ideal of a determinantal ideal "defined by shape" is given by its KRS image. We furthermore characterize those among these ideals that even have a Groebner basis of products of minors, and show that they can be characterized in terms of Greene's KRS invariants. Furthermore it is shown that for the ideal ge…

Groebner bases via linkage

In this paper, we give a sufficient condition for a set $\mathal G$ of polynomials to be a Gröbner basis with respect to a given term-order for the ideal $I$ that it generates. Our criterion depends on the linkage pattern of the ideal $I$ and of the ideal generated by the initial terms of the elements of $\mathcal G$. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix…

Improving DISPGB Algorithm Using the Discriminant Ideal

In 1992, V. Weispfenning proved the existence of Comprehensive Groebner Bases (CGB) and gave an algorithm to compute one. That algorithm was not very efficient and not canonical. Using his suggestions, A. Montes obtained in 2002 a more efficient algorithm (DISPGB) for Discussing Parametric Groebner Bases. Inspired in its philosophy, V. Weispfenning defined, in 2002, how to obtain a Canonical Comprehensive Groebner Basis (CCGB) for parametric polynomial ideals, and provided a constructive method. In this paper we use Weispfenning's CCGB ideas to make substantial improvements on Montes DISPGB al…

Groebner bases and combinatorics for binary codes

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.

在字典序Gröbner基上读取有理单变量表示

Reading Rational Univariate Representations on lexicographic Groebner bases

本文提出从二元消元理想的字典序Gröbner基中直接读取有理单变量表示（RUR）的闭式方法，无需理想处于形状位置，并基于FGLM核心给出简单算法，实验表明其与不认证参数化的先进实现竞争。

F5C: a variant of Faugere's F5 algorithm with reduced Groebner bases

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.

Signature-based algorithms to compute Groebner bases

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.

Universal Groebner Bases in Weyl Algebras

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.

Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties

We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by establishing (with a Groebner basis) a reduced and equidimensional limit whose Stanley-Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassman…

整数系数 Gröbner 基的模计算

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。

Oracle-supported drawing of the Groebner {\em escalier}

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…

Comprehensive Involutive Systems

In this paper, we consider parametric ideals and introduce a notion of comprehensive involutive system. This notion plays the same role in theory of involutive bases as the notion of comprehensive Groebner system in theory of Groebner bases. Given a parametric ideal, the space of parameters is decomposed into a finite set of cells. Each cell yields the corresponding involutive basis of the ideal for the values of parameters in that cell. Using the Gerdt-Blinkov algorithm for computing involutive bases and also the Montes algorithm for computing comprehensive Groebner systems, we present an alg…

Universal Groebner bases for maximal minors

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…

Hyperplane Sections, Groebner Bases, and Hough Transforms

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…

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

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…

关于2-行列式理想的Rees代数

On Rees algebras of 2-determinantal ideals

本文证明了2×n线性形式矩阵子式理想的Rees代数及其特殊纤维环具有Cohen-Macaulay和Koszul性质，通过分析Hilbert概形分层并结合Groebner退化揭示了其二次代数特征。

Groebner.jl：Julia 中的快速格罗贝格追踪

Groebner.jl: Fast Gröbner Tracing in Julia

研究如何加速格罗贝纳基计算中的重复运算，核心方法是用 Julia 实现 F4 算法并通过公共接口展示特拉弗索追踪法，支持 SIMD 友好系数类型，主要贡献是让 Julia 软件在相关应用中加速。

Computing Inhomogeneous Groebner Bases

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.

Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular syste…

A proof of Hilbert's Nullstellensatz based on Groebner bases

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.

Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.

Simple signature-based Groebner basis algorithm

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

Radical generic initial ideals

In this paper, we survey the theory of Cartwright-Sturmfels ideals. These are Z^n-graded ideals, whose multigraded generic initial ideal is radical. Cartwright-Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Groebner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright-Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determina…

New developments in the theory of Groebner bases and applications to formal verification

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 …

Deformations of Border Bases

Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to the degree form ideal works only under additional hypotheses, we introduce border basis schemes and universal border basis families. With their help the problem can be rephrased as the search for a certain rational curve on a border basis scheme. We construct the system of generators of the vanishing ideal of the border basis scheme in different wa…

Groebner and diagonal bases in Orlik-Solomon type algebras

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To …