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We propose a counterexample to the Grobner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V [ X , Y ] equipped with some irrational monomial order LT ( J ) generated by the leading terms of the elements of J is not finitely generated.

We introduce a new class of polynomial ideals associated to a simple graph, G. Let K [ E G ] be the polynomial ring on the edges of G and K [ V G ] the polynomial ring on the vertices of G. We associate to G an ideal, I ( X G ) , defined as the preimage of ( x i 2 − x j 2 : i , j ∈ V G ) ⊆ K [ V G ] by the map K [ E G ] → K [ V G ] which sends a variable, t e , associated to an edge e = { i , j } , to the product x i x j of the variables associated to its vertices. We show that K [ E G ] / I ( X G ) is a one-dimensional, Cohen–Macaulay, graded ring and we relate its degree with the number of vertices and connected components of the graph. We show that I ( X G ) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo–Mumford regularity of I ( X G ) providing the following sharp upper and lower bounds: μ ( G ) ≤ reg I ( X G ) ≤ | V G | − b 0 ( G ) + 1 , where μ ( G ) is the maximum vertex join number of the graph and b 0 ( G ) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Gr\"obner bases, factorization or sub-resultant computations., Comment: 15 pages

A set system F ⊆ 2 [ n ] shatters a given set S ⊆ [ n ] if 2 S = { F ∩ S : F ∈ F } . The Sauer-Shelah lemma states that in general, F shatters at least | F | sets. A set sytstem is called shattering-extremal if it shatters exactly | F | sets. In [Meszaros, T., “S-extremal set systems and Grobner bases”, Diploma Thesis, Budapest University of Technology and Economics, 2010] and [Meszaros, T., L. Ronyai, “Some combinatorial application of Grobner bases”, In: F. Winkler, Algebraic Informatics, CAI 2011, Lecture Notes in Computer Science 6742, Springer 2011, 64–83] an algebraic characterization of shattering-extremal set systems was given, which offered the possibility to generalize the notion of extremality to general finite vector systems. Here we generalize the results obtained for set systems to this more general setting, and as an application, strengthen a result of Dong, Li and Zhang from [Dong, T., Z. Li, S. Zhang, Finite sets of affine points with unique associated monomial order quotient bases, Journal of Algebra and its Applications 11(2) (2012), 1250025].

Let R ( h ) denote the polynomial ring in variables x 1 , … , x h over a specified field K. We consider all of these rings simultaneously, and in each use lexicographic (lex) monomial order with x 1 > ⋯ > x h . Given a fixed homogeneous ideal I in R ( h ) , for each d there is unique lex ideal generated in degree at most d whose Hilbert function agrees with the Hilbert function of I up to degree d. When we consider I R ( N ) for N ≥ h , the set B d ( I , N ) of minimal generators for this lex ideal in degree at most d may change, but B d ( I , N ) is constant for all N ≫ 0 . We let B d ( I ) denote the set of generators one obtains for all N ≫ 0 , and we let b d = b d ( I ) be its cardinality. The sequences b 1 , … , b d , … obtained in this way may grow very fast. Remarkably, even when I = ( x 1 2 , x 2 2 ) , one obtains a very interesting sequence, 0 , 2 , 3 , 4 , 6 , 12 , 924 , 409620 , … . This sequence is the same as H d − 1 + 1 for d ≥ 2 , where H d is the dth Hamilton number. The Hamilton numbers were studied by Hamilton and by Hammond and Sylvester because of their occurrence in a counting problem connected with the use of Tschirnhaus transformations in manipulating polynomial equations.

Given an arbitrary integer \(d>0\), we construct a homogeneous ideal I of the polynomial ring \(S = K[x_1, \ldots , x_{3d}]\) in 3d variables over a field K for which S/I is a Cohen–Macaulay ring of dimension d with the property that, for each of the integers \(0 \le r \le d\), there exists a monomial order \(

We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a $\mathbb{K}[X_1,\dots,X_r]$-module $\mathcal{M}$ of finite dimension $D$ as a $\mathbb{K}$-vector space, and given elements $f_1,\dots,f_m$ in $\mathcal{M}$, the problem is to compute syzygies between the $f_i$'s, that is, polynomials $(p_1,\dots,p_m)$ in $\mathbb{K}[X_1,\dots,X_r]^m$ such that $p_1 f_1 + \dots + p_m f_m = 0$ in $\mathcal{M}$. Assuming that the multiplication matrices of the $r$ variables with respect to some basis of $\mathcal{M}$ are known, we give an algorithm which computes the reduced Gr\"obner basis of the module of these syzygies, for any monomial order, using $O(m D^{\omega-1} + r D^\omega \log(D))$ operations in the base field $\mathbb{K}$, where $\omega$ is the exponent of matrix multiplication. Furthermore, assuming that $\mathcal{M}$ is itself given as $\mathcal{M} = \mathbb{K}[X_1,\dots,X_r]^n/\mathcal{N}$, under some assumptions on $\mathcal{N}$ we show that these multiplication matrices can be computed from a Gr\"obner basis of $\mathcal{N}$ within the same complexity bound. In particular, taking $n=1$, $m=1$ and $f_1=1$ in $\mathcal{M}$, this yields a change of monomial order algorithm along the lines of the FGLM algorithm with a complexity bound which is sub-cubic in $D$., Comment: 34 pages, 7 algorithms, Journal of Complexity

Bayer-Stillman showed that $reg(I) = reg(gin_\tau(I))$ when $\tau$ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order $\tau$ satisfying $reg(I) = reg(gin_\tau(I))$ for all ideals $I$. We also show that if $gin_{\tau_1}(I) = gin_{\tau_2}(I)$ for all $I$, then $\tau_1 = \tau_2$., Comment: 10 pages

In this paper taking a question in [1] into background we investigate the uniqueness of a non-constant polynomial with the differential monomial generated by a non-constant mermorphic function f. Our result will also extend a result of Banerjee-Majumder[2] given earlier. An open question is also posed, in the paper, for future investigation.

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 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.

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.

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.

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

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).

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 /

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.

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

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

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.

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.

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.

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.

Heinzer, Mirbagheri, Ratliff, and Shah investigate parametric decomposition of monomial ideals on a regular sequence of a commutative ring R with identity 1 and prove that if every finite intersection of monomial ideals in R is again a monomial ideal, then each monomial ideal has a unique irredundant parametric decomposition. Sturmfels, Trung, and Vogels prove a similar result without the uniqueness. Bayer, Peeva, and Strumfels study generic monomial ideals, that is monomial ideals in the polynomial ring such that no variable appears with the same nonzero exponent in two different minimal generators, and for these ideals they prove the uniqueness of the irredundant irreducible decompositions and give an algorithm to construct this unique irredundant irreducible decomposition. In this paper, we present three algorithms for finding the unique irredundant irreducible decomposition of any monomial ideal.

Among all the restrictions of weight orders to the subsets of monomials with a fixed degree, we consider those that yield a total order. Furthermore, we assume that each weight vector consists of an increasing tuple of weights. Every restriction, which is shown to be achieved by some monomial order, is interpreted as a suitable linearization of the poset arising by the intersection of all the weight orders. In the case of three variables, an enumeration is provided. For a higher number of variables, we show a necessary condition for obtaining such restrictions, using deducibility rules applied to homogeneous inequalities. The logarithmic version of this approach is deeply related to classical results of Farkas type, on systems of linear inequalities. Finally, we analyze the linearizations determined by sequences of prime numbers and provide some connections with topics in arithmetic.

Let ( f 1 , … , f s ) ∈ Q p [ X 1 , … , X n ] s be a sequence of homogeneous polynomials with p-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since Q p is not an effective field, classical algorithm does not apply. We provide a definition for an approximate Grobner basis with respect to a monomial order w. We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals 〈 f 1 , … , f i 〉 are weakly-w-ideals. The conjecture of Moreno-Socias states that for the grevlex ordering, such sequences are generic. Two variants of that strategy are available, depending on whether one leans more on precision or time-complexity. For the analysis of these algorithms, we study the loss of precision of the Gauss row-echelon algorithm, and apply it to an adapted Matrix-F5 algorithm. Numerical examples are provided. Moreover, the fact that under such hypotheses, Grobner bases can be computed stably has many applications. Firstly, the mapping sending ( f 1 , … , f s ) to the reduced Grobner basis of the ideal they span is differentiable, and its differential can be given explicitly. Secondly, these hypotheses allow to perform lifting on the Grobner bases, from Z / p k Z to Z / p k + k ′ Z or Z . Finally, asking for the same hypotheses on the highest-degree homogeneous components of the entry polynomials allows to extend our strategy to the affine case.

In this paper we consider a free associative algebra on three generators over an arbitrary fieldK. Given a term ordering on the commutative polynomial ring on three variables overK, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in non-commutative Gröbner basis theory.

It is proved that iteration of the process of adjoining an identity to a semigroup gives birth naturally to the lexicographical ordering on the additive semigroups of n-tuples of nonnegative integers.

LetS=k[x1,…, xn] be a polynomial ring over a fieldk, Ia monomial ideal ofS, R=S/I. We construct a resolution ofkoverR.

Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with gin_0(I), or with Gin_0(I) whenever the order is the reverse-lexicographic order. When the characteristic of K is zero, a zero-generic initial ideal is the usual generic initial ideal. We show that Gin_0(I) is endowed with many interesting properties, some of which are easily seen, e.g., it is a strongly stable monomial ideal in any characteristic, and has the same Hilbert series as I; some other properties are less obvious: it shares with I the same Castelnuovo-Mumford regularity, projective dimension and extremal Betti numbers. Quite surprisingly, gin_0(I) also satisfies Green's Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0. We also prove the analogue for local cohomology of Pardue's Conjecture for Betti numbers: we show that the Hilbert functions of local cohomology modules of a quotient ring of a polynomial ring modulo a weakly stable ideal are independent of the characteristic of the coefficients field., Comment: Added a generalization of the main result of arXiv:1307.2707, "Minimal Castelnuovo-Mumford regularity fixing the Hilbert polynomial", by F. Cioffi et al. to any characteristic. Some minor changes

In this paper, we consider monomial ideals in k [ x 1 , x 2 , x 3 ]. For each d ≥ 1, we compute the smallest number of monic monomials of degree d − 1 such that the ideal which they generate contains all monomials of degree d . This answers, for the case n = 3, a question posed by A.V. Geramita, D. Gregory, and L. Roberts.

Let S=K[x1,…,xn] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.

As a generalization of the facet ideal of a forest, we define monomial ideal of forest type and show that monomial ideals of forest type are pretty clean. As a consequence, we show that if I is a monomial ideal of forest type in the polynomial ring S, then Stanley's decomposition conjecture holds for S/I. The other main result of this article shows that a clutter is totally balanced if and only if it has the free vertex property, and which is also equivalent to say that its edge ideal is a monomial ideal of forest type or is generated by an M sequence.

Let IF be a field, V ⊆ IFn be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V. To study these, one often considers I(V), the vanishing ideal of V in the polynomial ring IF[x1,..., xn]. Grobner bases and standard monomials of I(V) appear to be useful in this context, leading to structural results on V. Here we survey some work of this type. At the end of the paper a new application of this kind is presented: an algebraic characterization of shattering-extremal families and a fast algorithm to recognize them.

The quotient bases for zero-dimensional ideals are often of interest in the investigation of multivariate polynomial interpolation, algebraic coding theory, and computational molecular biology, etc. In this paper, we discuss the properties of zero-dimensional ideals with unique monomial order quotient bases, and verify that the vanishing ideals of Cartesian sets have unique monomial order quotient bases. Furthermore, we reveal the relation between Cartesian sets and the point sets with unique associated monomial order quotient bases.

In the present paper we consider a class of involutive monomial divisions pairwise constructed by the partition of variables into multiplicative and nonmultiplicative generated by a total monomial ordering. If this ordering is admissible or the inverse of an admissible ordering, then the involutive division generated possesses all algorithmically important properties such as continuity, constructivity, and noetherianity. Among all such divisions, we single out those generated by antigraded monomial orderings. We demonstrate, by example of the antigraded lexicographic ordering, that the divisions of this class are heuristically better than the classical Janet division. The last division is pairwise generated by the pure lexicographic ordering and up to now has been considered as computationally best.

We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gr\"obner basis. We then obtain an analogous result for lattice path matroids., Comment: 9 pages, 4 figures