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We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.

In this paper, we address the following question: for a nonzero finitely generated ideal [Formula: see text] of a multivariate polynomial ring [Formula: see text] over a coherent ring [Formula: see text], fixing a monomial order [Formula: see text] on [Formula: see text], is the trailing terms ideal [Formula: see text] of [Formula: see text] (that is, the ideal generated by the trailing terms of the nonzero polynomials in [Formula: see text]) finitely generated? We show that while [Formula: see text] can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances).

In this chapter we consider classes of binomial ideals which are naturally attached to finite simple graphs. The first of these classes are the binomial edge ideals. These ideals may also be viewed as ideals generated by a subset of 2-minors of a (2 × n)-matrix of indeterminates. Their Grobner bases will be computed. Graphs whose binomial edge ideals have a quadratic Grobner basis are called closed graphs. A full classification of closed graphs is given. For an arbitrary graph the initial ideal of the binomial edge ideal (for a suitable monomial order) is a squarefree monomial ideal. This has the pleasant consequence that the binomial edge ideal itself is a radical ideal. Its minimal prime ideals are determined in terms of cut point properties of the underlying graph. Based on this information, the closed graphs whose binomial edge ideal is Cohen–Macaulay are classified. In the subsequent sections, the resolution of binomial edge ideals is considered and a bound for the Castelnuovo–Mumford regularity of these ideals is given. Finally, the Koszul property of binomial edge ideals is studied. Intimately related to binomial edge ideals are permanental edge ideals and Lovasz, Saks, and Schrijver edge ideals. Their primary decomposition will be studied.

The purpose of Chapter 1 is to provide the reader with sufficient knowledge of the basic theory of Grobner bases which is required for reading the later chapters. In Section 1.1, we study Dickson’s Lemma, which is a classical result in combinatorics. Grobner bases are then introduced and Hilbert’s Basis Theorem and Macaulay’s Theorem follow. In Section 1.2, the division algorithm, which is the framework of Grobner bases, is discussed with a focus on the importance of the remainder when performing division. The highlights of the fundamental theory of Grobner bases are Buchberger’s criterion and Buchberger’s algorithm presented in Section 1.3. Furthermore, in Section 1.4, elimination theory will be introduced. This theory is very useful for solving a system of polynomial equations. Finally, in Section 1.5, we discuss the universal Grobner basis of an ideal. This is a finite set of polynomials which is a Grobner basis for the ideal with respect to any monomial order.

This paper studies the minimal monomial basis of the n-variable Birkhoff interpolation problem. First, the authors give a fast B-Lex algorithm which has an explicit geometric interpretation to compute the minimal monomial interpolation basis under lexicographic order and the algorithm is in fact a generalization of lex game algorithm. In practice, people usually desire the lowest degree interpolation polynomial, so the interpolation problems need to be solved under, for example, graded monomial order instead of lexicographic order. However, there barely exist fast algorithms for the nonlexicographic order problem. Hence, the authors in addition provide a criterion to determine whether an n-variable Birkhoff interpolation problem has unique minimal monomial basis, which means it owns the same minimal monomial basis w.r.t. arbitrary monomial order. Thus, for problems in this case, the authors can easily get the minimal monomial basis with little computation cost w.r.t. arbitrary monomial order by using our fast B-Lex algorithm.

An algebraic method is proposed for selective harmonic elimination PWM (SHEPWM). By computing its Groebner bases under the pure lexicographic monomial order, the nonlinear high-order SHE equations are converted to an equivalent triangular form, and then a recursive algorithm is used to solve the triangular equations one by one. Based on the proposed method, a user-friendly software package has been developed and some computation results are given. Unlike the commonly used numerical and intelligent methods, this method does not need to choose the initial values and can find all the solutions. Also, this method can give a definite answer to the question of whether the SHE equations have solutions or not, and the accuracy of the solved switching angles are much higher than that of the reference method. Compared with the existing algebraic methods, such as the resultant elimination method, the calculation efficiency is improved. Experimental verification is also shown in this paper.

Let I ⊂ K[x 1,…, x n ] be an ideal and G be the reduced Grobner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x 1,…, x n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm.

In this article we study a class of orders called {\it monomial orders} in a central simple algebra over a non-Archimedean local field. Monomial orders are easily represented and they may be also viewed as a direct generalization of Eichler orders in quaternion algebras. A criterion for monomial orders to be Gorenstein or to be Bass is given. It is shown that a monomial order is Bass if and only if it is either a hereditary or an Eichler order of period two., 13 pages; fix typos in the proof of Theorem 3.1

For a given node set Z and the corresponding interpolation conditions, one of the key problems in multivariate Birkhoff interpolation is determining a monomial basis that spans the interpolation space. However, a slight perturbation of the node set will result in a different monomial basis. We propose a more general multivariate Birkhoff interpolation scheme and provide a numerical algorithm that can determine the stable monomial basis to ensure that the interpolation polynomial always exists in the space spanned by the monomial basis for any perturbation of the node set within a given error e .

We study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We focus on monomial properties expressed in terms of the number of distinct variables occurring in a monomial. Our first result is a randomized FPT algorithm for detection of a monomial having at least k distinct variables, parametrized with respect to k. For a more restricted class of circuits, we can also provide a deterministic FPT algorithm for detection of a monomial having at most k distinct variables parametrized by the degree of the polynomial represented by the input circuit. Furthermore, we derive several hardness results on detection of monomials with such properties within exact, parametrized and approximation complexity. In particular, we observe that the detection of a monomial having at most k distinct variables is W 2 -hard for the parameter k. An O * ( 2 k ) -time detection of a monomial with ?k variables in a polynomial given by a circuit.An O * ( 2 l ) -time detection of a monomial with ?k variables in a restricted l-degree polynomial.Detecting a monomial with ?k distinct variables is W 2 -hard with respect to k.Finding min ? k s.t. there is a monomial with ?k variables is inapproximable within O ( 2 log 1 - ? ? n ) .Finding max ? k s.t. there is a monomial with ?k variables is inapproximable within 1 - 1 e + ? .

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotient with respect to a monomial order. We also provide a lower bound for some Betti numbers of powers of a square-free monomial ideal which is generated in a single degree., arXiv admin note: text overlap with arXiv:math/0307222 by other authors

In this paper we focus on the algorithms and their implementations for decomposing an arbitrary polynomial set simultaneously into pairs of lexicographic Grobner bases and normal triangular sets with inherent connections in between and associated zero relationship with the polynomial set. In particular, a method by temporarily changing the variable orderings to handle the failure of the variable ordering assumption is proposed to ensure splitting needed for characteristic decomposition. Experimental results of our implementations for (strong) characteristic decomposition with comparisons with available implementations of triangular decomposition are also reported.

For a quantized enveloping algebra of finite type, one can associate a natural monomial to a dominant weight. We show that these monomials for types A5 and D4 are semitight (i.e., a ℤ-linear combination of elements in the canonical basis) by a direct calculation.

We prove that a valuation domain \(\mathbf{V}\) has Krull dimension \(\le \)1 if and only if, for any \(n\), fixing the lexicographic order as monomial order in \(\mathbf{V}[X_1,\ldots ,X_n]\), for every finitely generated ideal \(I\) of \(\mathbf{V}[X_1,\ldots ,X_n]\), the ideal generated by the leading terms of the elements of \(I\) is also finitely generated. This proves the Grobner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prufer domains. As a “scoop”, contrary to the common idea that Grobner bases can be computed exclusively on Noetherian ground, we prove that computing Grobner bases over \(\mathbf{R}[X_1,\ldots , X_n]\), where \(\mathbf{R}\) is a Prufer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of \(\mathbf{R}\) is \(\le \)1.

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone., 24 pages; corrected typos, added references and an acknowledgment. The paper is accepted for publication in Mathematics of Computation

Let R = K[x, y, z] denote the polynomial ring in three variables over an arbitrary field K. We study the factorial closure B(E) of certain R-modules E of projective dimension 1, called monomial modules. The module E is—in a certain sense—related to the defining ideal of a monomial affine space curves in In general, we prove that B(E) is different from Sym R (E). This extends a particular case of an R-module E with this property studied by Samuel [7]. As the main result, we characterize those monomial R-modules E such that B(E) is an R-algebra of finite type and generated in degree 1 and 2 over the symmetric algebra Sym R (E).

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

We say that a rational map on P n is a monomial map if it can be expressed in some coordinate system as [ F 0 : ⋯ : F n ] where each F i is a monomial. We consider arithmetic dynamics of monomial maps on P 2 . In particular, as Silverman (1993) explored for rational maps on P 1 , we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on P 2 is a polynomial, then the first such iterate is 1, 2, 3, 4, 6, 8, or 12. We then completely determine all monomial maps whose orbits always contain just finitely many integral points. Our condition is based on the exponents in the monomials. In cases when there are finitely many integral points in all orbits, we also show that the sizes of the numerators and the denominators are comparable. The main ingredients of the proofs are linear algebra, such as Perron–Frobenius theorem.

Taketa shows that all monomial groups (commonly written as M -groups) are solvable. Gajendragadkar gives the notion of π-factorable character. We show that an irreducible character of an M-group is primitive if it is π-factorable. Issacs proves that product of two monomial characters is a monomial. We extend this fact to include any finite number of monomial characters consequently we prove that any product of finite number of M-groups is an M-group. We show that any group of order 45 is an M-group and for any group G, the factor group /

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 obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip., 15 pages, 1 table, 2 algorithms (corrected version with new Prop. 3.8 and proof); Journal of Symbolic Computation 46 (2011)

Let I1 ⊂ I2 ⊂ . . . be an increasing sequence of ideals of the ring Z[X], X = (x1, . . . , xn), and let I be their union. We propose an algorithm to compute the Gr¨obner base of I under the assumption that the Gr¨obner bases of the ideal QI of the ring Q[X] and of the ideals I ⊗ (Z/mZ) of the rings (Z/mZ)[X] are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.

Grobner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field $$ \mathbb{F}_2 $$ . In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.

A matrix is said to be monomial if every row and column has only one nonzero entry. Let G be a group. A representation ρ: G → GL n (ℂ) is said to be a monomial representation of G if there exists a basis with respect to which ρ(g) is a monomial matrix for every g ∈ G. We use elementary methods to classify the irreducible monomial representations of the groups L 2(q), L 3(q) and their natural decorations.

In this paper, we answer a question posed by Y.H. Shen. We prove that if I is an m-generated squarefree monomial ideal in the polynomial ring S=K[x1,…,xn] with K a field, then sdepthI⩾n−⌊m/2⌋. The proof is inductive and uses the correspondence between a Stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by Herzog, Vladoiu and Zheng.

The goal of this work is to analyze various classes of finite and total monomial orderings. The concept of monomial ordering plays the key role in the theory of Grobner bases: every basis is determined by a certain ordering. At the same time, in order to define a Grobner basis, it is not necessary to know ordering of all monomials. Instead, it is sufficient to know only a finite interval of the given ordering. We consider combinatorics of finite monomial orderings and its relationship with cells of a universal Grobner basis. For every considered class of orderings (weakly admissible, convex, and admissible), an algorithm for enumerating finite orderings is discussed and combinatorial integer sequences are obtained. An algorithm for computing all minimal finite orderings for an arbitrary Grobner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Grobner basis for an arbitrary zero-dimensional ideal.

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid-1980s and developed a framework for performing such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x,y) that are suitable for this framework. We then perform such computations for ideals in the polynomial ring K[x,y]. Interestingly, for these valuations, some ideals have finite Gröbner bases with respect to a valuation that are not Gröbner bases with respect to any monomial order, whereas other ideals only have Gröbner bases that are infinite.

We give an explicit construction of Grobner-Shirshov pairs and monomial bases for finite-dimensional irreducible representations of the simple Lie algebra . We also identify the monomial basis consisting of the reduced monomials with a set of semistandard tableaux of a given shape, on which we give a colored oriented graph structure.

In this article, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Grobner–Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.

The Dixon resultant is proposed as an alternative to Grobner basis or multipolynomial resultant approaches for solving systems of polynomial equations inherent in geodesy. Its smallness in size, high density (ratio on the number of nonzero elements to the number of all elements), speed, and robustness (insensitive to combinatorial sequence and monomial order, e.g., Grobner basis) makes it extremely attractive compared to its competitors. Using 3D-intersection and conformal C 7 datum transformation problems, we compare its performance to those of the Sturmfels’s resultant and Grobner basis. For the 3D-intersection problem, Sturmfels’s resultant needed 0.578 s to solve a 6 × 6 resultant matrix whose density was 0.639, the Dixon resultant on the other hand took 0.266 s to solve a 4 × 4 resultant matrix whose density was 0.870. For the conformal C 7 datum transformation problem, the Dixon resultant took 2.25 s to compute a quartic polynomial in scale parameter whereas the computaton of the Grobner basis fails. Using relative coordinates to compute the quartic polynomial in scale parameter, the Grobner basis needed 0.484 s, while the Dixon resultant took 0.016 s. This highlights the robustness of the Dixon resultant (i.e., the capability to use both absolute and relative coordinates with any order of variables) as opposed to Grobner basis, which only worked well with relative coordinates, and was sensitive to the combinatorial sequence and order of variables. Geodetic users uncomfortable with lengthy expressions of Grobner basis or multipolynomial resultants, and who aspire to optimize on the attractive features of Dixon resultant, may find it useful.

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of $I$ grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials., Fixed an argument in the proof; added some comments

This article focuses on two recent results on multiplicative invariants of finite reflection groups: Lorenz (2001) showed that such invariants are affine normal semigroup algebras, and Reichstein (2003) proved that the invariants have a finite SAGBI basis. Reichstein (2003) also showed that, conversely, if the multiplicative invariant algebra of a finite group G has a SAGBI basis, then G acts as a reflection group. There is no obvious connection between these two results. We will show that multiplicative invariants of finite reflection groups have a certain embedding property that implies both results simultaneously.

We give a simple algorithm to decide whether a monomial ideal of nite colength in a polynomial ring is licci, i.e., in the linkage class of a complete intersection. The algorithm proves that whether or not such an ideal is licci does not depend on whether we restrict the linkage by only allowing monomial regular sequences, or homogeneous regular sequences, or arbitrary regular sequences. We apply our results on monomial ideals to compare when an ideal is licci versus when its initial ideal in some term order is licci. We also apply an idea of Migliore and Nagel to prove that monomial ideals of nite colength are always glicci, i.e., in the Gorenstein linkage class of a complete intersection. However, our proof requires the use of non-homogeneous Gorenstein links.

We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair $(S,>)$ consisting of a birational sequence and a monomial order, we attach to the affine variety $G/\hskip -3.5pt/U$ a monoid $\Gamma=\Gamma(S,>)$. As a side effect we get a vector space basis $\mathbb B_{\Gamma}$ of $\mathbb C[G/\hskip -3.5pt/U]$, the elements being indexed by $\Gamma$. The basis $\mathbb B_{\Gamma}$ has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion \cite{AB} to this more general setting, once one knows that the monoid $\Gamma$ is finitely generated and saturated., Comment: 35 pages, to appear in Adv. Math

In this paper, an algebraic method which is based on the groebner bases theory is proposed to solve the selective harmonic elimination PWM(SHEPWM) problem. Firstly, the SHEPWM equations are transformed to polynomial equations by using the multiple-angle formulas and variables substitution, then, the polynomial equations are converted to an equivalent triangular form by computing the reduced groebner bases under the pure lexicographic monomial order, finally, the triangular equations can be solved by a successive back-substitution manner just like the Gaussian elimination which is used to solve the linear equations. A software package is developed under the symbolic computing software Maple, and it can solve the SHEPWM equations with no more than 8 switching angles for the single-phase inverters and 5 switching angles for the three-phase inverters. In contrast with the common used numerical and intelligent methods, the advantages of this method are there is no need to choose the initial values and all the real solutions can be found. Experimental results verify the correctness of the switching angles computed by the proposed method.

In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gr\"obner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree. Let $k$ be a finite field of cardinality $q^n$ and let $k'$ be its subfield of cardinality $q$. Let $\mathcal{F} \subset k[X_0,\ldots,X_{m-1}]$ be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of $\mathcal{F}$, which depends on $q$, $m$, the last fall degree of $\mathcal{F}$, the degree of $\mathcal{F}$ and the number of solutions of $\mathcal{F}$, but not on $n$. This shows that such Weil descent systems can be solved efficiently if $n$ grows. In particular, we apply these results for multi-HFE and essentially show that multi-HFE is insecure. Finally, we discuss that the degree of regularity (or last fall degree) of Weil descent systems coming from summation polynomials to solve the elliptic curve discrete logarithm problem might depend on $n$, since such systems without field equations are not zero-dimensional., Comment: 16 pages, changed definition of tau and revised Section 5

s-sequences and d-sequences are fundamental sequences in- tensively studied in many fields of algebra. In this paper we are interested in dealing with monomial sequences associated to graphs in order to es- tablish conditions for which they are s-sequences and/or d-sequences. In this work we consider monomial sequences establishing conditions for which they are s-sequences and d-sequences in order to deduce some properties of the symmetric algebra of monomial ideals, in particular of some monomial ideals arising from graphs. In (2) the notion of d-sequence was firstly given by Huneke for the study of Rees rings. In (1) the notion of s-sequence is employed to compute the invariants of the symmetric algebra of finitely generated modules and it is proved that any d-sequence is a strong s-sequence. Some properties about monomial s-sequences are studied. In (1) it is given a necessary and sufficient condition for monomial sequences of length three to be s-sequences. Afterwards, in (6) it is shown a necessary and sufficient condition for monomial squarefree sequences of length four, and in (5) a more general statement for monomial sequences of any length related to forests. Our aim is to find necessary and sufficient conditions for monomial sequences of length greater than four to be s-sequences. Some results are obtained for edge sequences associated to very important classes of graphs and components of them. The d-sequences have been intensively studied by many algebraists, the monomial d-sequences are characterized in (6). Our aim is to investigate the notion of d-sequence for the monomial ideals arising from simple graphs in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs.

In §10 of Chapter 2 we discussed some criteria designed to identify situations where it is possible to see in advance that an S-polynomial remainder will be zero in Buchberger’s algorithm. Those unnecessary S-polynomial remainder calculations are in fact the main computational bottleneck for the basic form of the algorithm. Finding ways to avoid them, or alternatively to replace them with less expensive computations, is the key to improving the efficiency of Grobner basis calculation. The algorithms we discuss in this chapter apply several different approaches to achieve greater efficiency. Some of them use Grobner bases of homogeneous ideals or ideas inspired by the special properties of Grobner bases in that case. So we begin in §1 by showing that the computation of a homogeneous Grobner basis can be organized to proceed degree by degree. This gives the framework for Traverso’s Hilbert driven Buchberger algorithm, discussed in §2, which uses the Hilbert function of a homogeneous ideal to control the computation and bypass many unnecessary S-polynomial remainder calculations. We also show in §1 that the information generated by several S-polynomial remainder computations can be obtained simultaneously via row operations on a suitable matrix. This connection with linear algebra is the basis for Faugere’s F4 algorithm presented in §3. Finally, we introduce the main ideas behind signature-based Grobner basis algorithms, including Faugere’s F5 algorithm, in §4.

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.

We give a combinatorial description of the roots of the Bernstein–Sato polynomial of a monomial ideal using the Newton polyhedron and some semigroups associated to the ideal.

In this article, we give a new realization of crystal bases for irreducible highest weight modules over U q (G 2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Communicated by K. Misra

We develop a new general method for constructing a polynomial ideal that has the same radical as a given monomial ideal, but has fewer generators. This provides upper bounds for the arithmetical rank of monomial ideals. An explicit formula is given for ideals generated by squarefree monomials of degree 2.

Let R := k[x1, x2, x3] be a 3-dimensional polynomial ring over a field k. We prove that a (x1, x2, x3)-primary monomial ideal in such a ring is *-simple if it has minimal multiplicity with respect to each of its fibers. We also survey the known classes of *-simple monomial ideals in rings of this type and show that this new class is distinct from other known classes.

This paper extends the theory of the Gröbner fan and Gröbner walk for ideals in polynomial rings to the case of submodules of free modules over a polynomial ring. The Gröbner fan for a submodule creates a correspondence between a pair consisting of a cone in the fan and a point in the support of the cone and a pair consisting of a leading monomial submodule (or equivalently, a reduced marked Gröbner basis) and a grading of the free module over the ring that is compatible with the choice of leading monomials. The Gröbner walk is an algorithm based on the Gröbner fan that converts a given Gröbner basis to a Gröbner basis with respect to a different monomial order; the point being that the Gröbner walk can be more efficient than the standard algorithms for Gröbner basis computations with difficult monomial orders. Algorithms for generating the Gröbner fan and for the Gröbner walk are given.

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute upper and lower bounds on the maximal order type., 40 pages

Parks [7] showed that there is an one to one correspondence between good pairs of subgroups in G and irreducible monomial characters of G. This provides a useful criterion for a group to be monomial. In this paper, we study relative monomial groups by defining triples in G, and find relationships between the triples and irreducible relative monomial characters.

A trick introduced by Macaulay, in order to blow up a monomial ideal into a point ideal whose maximal ideal is the given monomia ideal, is sligtly generalized to allow to write down a Grobner basis whose maximal ideal is a given monomial ideal.

One of methods to solve integer programming problems consists in the method which uses groebner bases. An arbitrary solution for a state equation Ax=b(A∈Zmxn,b∈Zmx1) of Petri nets means a firing count vector. Then finding a nonnegative integer solution x∈Znx1+ for Ax=b in Petri nets is one of integer programming problems. In this paper, the method to obtain generators of solutions in Petri nets by using groebner bases is proposed and investigated. moreover, Petri nets have an ill property that the number of minimal support T-invariants increases in exponential when places and transitions are increased. Then, the number of groebner bases and calculation time of groebner bases are measured by using a symbolic computation system or a computer algebra system; Maple 7.

We describe (reduced) Grobner bases of the ideal of polynomials over a field, which vanish on the set of characterisic vectors of the complete unifom families ({\scriptstyle [n]}\\[-4pt]{\scriptstyle d}). An interesting feature of the results is that they are largely independent of the monomial order selected. The bases depend only on the ordering of the variables. We can thus use past results related to the lex order in the presence of degree-compatible orders, such as deglex. As applications, we give simple proofs of some known results on incidence matrices.

1. Since the introduction by Weispfenning [We92] of Comprehensive Grobner bases and systems (CGB and CGS), different algorithms have been developed for obtaining a CGS. Let K be a computable field and K an algebraically closed extension of K. Given a generating set {p1, · · · , ps} of the parametric ideal I ⊂ K[a][x], where a = a1, . . . , am are the parameters and x = x1, . . . , xn the variables, and a monomial order x in the variables, a CGS is a set of pairs (Si, Bi) with Si ⊂ K (called segments) and Bi ⊂ K[a][x] (called bases) that specialize to a Grobner basis in K[x] for all points a ∈ Si. Probably the fastest family of algorithms are based [Ka97] on the stability of Grobner bases in K[x, a] with a product order with x a by specialization. Sato and Suzuki [SuSa06], Nabeshima [Na07] and Kapur-Sun-Wang [KaSuWa10] have successively improved the most popular of these algorithms. The objective of them is to obtain a CGS without any more requirements, and they can benefit from the existing fast implementations of ordinary Grobner bases computations. Moreover, the last version in [KaSuWa10] already obtains a disjoint CGS. 2. Another family of algorithms [Mo02, We03, MaMo09], have as objective obtaining a canonical CGS with good properties for applications. These algorithms work in K[a][x] and use only the x as variables, but on the counterpart they need specific algorithms for Grobner bases computations. Based on Wibmer’s Theorem [Wi07] we have introduced the canonical Grobner cover [MoWi10] and implemented it in the Singular grobcov.lib library whose beta version and an executable tutorial can be downloaded at [Mo-web]. The canonical Grobner cover consist of a set of triplets {(lpp1, B1, S1), . . . , (lppr, Br, Sr)} with the following properties