Ibrahim Al-Ayyoub
In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).
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In this article we compute a minimal Groebner basis for the symmetric algebra for certain affine Monomial Curves, as an R-module. Keywords: Monomial Curves, Groebner Basis, Symmetric Algebra. Mathematics Subject Classification 2000: 13P10, 13A30 .
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.
Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on rational function fields of two variables that are suitable for this framework. For these valuations, we explicitly compute the image of an underlying polynomial ring and use this to perform computations concerning ideals in the polynomial ring. Interestingly, for these valuations, some ideals have a fini…
Let $I\subseteq S=K[x_1,\ldots,x_n]$ be a homogeneous ideal equipped with a monomial order $<$. We show that if $\operatorname{in}_<(I)$ is a square-free monomial ideal, then $S/I$ and $S/\operatorname{in}_<(I)$ have the same connectedness dimension. We also show that graphs related to connectedness of these quotient rings have the same number of components. We also provide consequences regarding Lyubeznik numbers. We obtain these results by furthering the study of connectedness modulo a parameter in a local ring.
We determine a Groebner basis for the secant ideal of the toric ideal associated to the second hypersimplex, with respect to any circular term order. The Groebner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Groebner degenerations for secant ideals, combinatorial characterizations of the secant ideals of monomial ideals, and the relations between secant ideals and prolongations.
This paper introduces a strategy for signature-based algorithms to compute Groebner basis. The signature-based algorithms generate S-pairs instead of S-polynomials, and use s-reduction instead of the usual reduction used in the Buchberger algorithm. There are two strategies for s-reduction: one is the only-top reduction strategy which is the way that only leading monomials are s-reduced. The other is the full reduction strategy which is the way that all monomials are s-reduced. A new strategy, which we call selective-full strategy, for s-reduction of S-pairs is introduced in this paper. In the…
The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Groebner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp-Massey-like decoding algorithm.
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.
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…
Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Groebner basis in a setting where a_monoid_ acts by_homomorphisms_ on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive…
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.
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.
In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.
Groebner.jl: A package for Gröbner bases computations in Julia
本文提出Julia包Groebner.jl，基于F4算法计算Gröbner基，支持模素数整数与有理数运算及多线程，结合符号计算技术与Julia类型系统，在性能与可扩展性上达到或超越现有最先进水平。
Given a finite set $\mathbf{X}$ of distinct points, Marinari-Mora's 'Axis of Evil Theorem' states that a combinatorial algorithm and interpolation enable to find a 'linear' factorization for a lexicographical minimal Groebner basis $\mathcal{G}(I(\mathbf{X}))$ of the zerodimensional radical ideal $I(\mathbf{X})$. In this work we provide such algorithm, showing that it ends in a finite number of steps and that it actually provides the correct result. The 'Axis of Evil' algorithm takes as input the monomial basis of the initial ideal $T(I(\mathbf{X}))$ but its starting point is the (finite) Groe…
We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The ob…
The theory of Groebner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Groebner Basis, and indeed in finding alternative methods of computing Groebner Bases. Two of these methods include the Groebner Walk method and the computation of Involutive Bases. By the mid 1980's, Buchberger's work had been generalised for noncommutative polynomial rings by Bergman and Mora. This thesis provides the corresponding generalisati…
For the ideal $\mathfrak{p}$ in $k[x, y, z]$ defining a space monomial curve, we show that $\mathfrak{p}^{(2 n - 1)} \subseteq \mathfrak{m} \mathfrak{p}^{n}$ for some positive integer $n$, where $\mathfrak{m}$ is the maximal ideal $(x, y, z)$. Moreover, the smallest such $n$ is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that $\mathfrak{p}^{(3)} \subseteq \mathfrak{m} \mathfrak{p}^2$ if $k$ is a field of characteristic not $3$; however, the stable Harbourne conjecture holds for space monomial curves as they claimed.
The purpose of this paper is twofold. 1. We give combinatorial bounds on the ranks of the groups $\Tor^{R}_\bullet(k,k)_\bullet$ in the case where $R = k[Λ]$ is an affine semi-group ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show that if the bounds hold for a field $k$ then they hold for $\field[Λ]$ and any field $\field$. Moreover, we provide a combinatorial construction for a free resolution of $\field$ over $\field[Λ]$ which achieves these bounds. 2. We extend the lexicographic discrete Morse fu…
Starting from \cite{Ayy2} we compute the Groebner basis for the defining ideal, P, of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P, inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combin…
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…
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…
Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Groebner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications. In this paper we compare Groebner basis schemes with border basis schemes. It is…
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…
In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.
Let $C$ be a Gorenstein non complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the minimal number of generators of the tangent cone of $C$. Special attention will be paid to the case where $C$ has Cohen-Macaulay tangent cone at the origin.
Given a finite alphabet X and an ordering on the letters, the map σsends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal <σ(I)> generated by σ(I) in the free monoid is finitely generated. Whether there exists an ordering such that <σ(I)> is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.
Construction of Involutive Monomial Sets for Different Involutive Divisions
本文研究不同对合除法下单项式集的对合完备化，提出每种除法对应独立完备化过程，多项式情形产生冗余对合基，并比较了Mathematica与C实现的Janet除法性能。
In this article, we find the equations defining the Rees algebra for certain Monomial Curves explicitly and use them to prove that the blowup scheme is not smooth. This proves a conjecture of Francia in affirmative, which says that a dimension one prime in a regular local ring is a complete intersection if it has a smooth blowup.
We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation type at most $n-2$ as long as $n \leq 5$. In general, we establish the defining equations of the Rees algebra in this case. Furthermore, we give a class of examples with relation type at least $n-2$, where $n=μ(I)$. We also provide new classes of ideals of linear type. We propose the construction of a graph, namely the generator graph of an ideal, where the mon…
We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to deduce some finiteness results about leading monomial ideals of any fixed ideal, namely: (1) the number of minimal leading monomial ideals with respect to total orderings is finite; (2) the number of leading monomial ideals with respect to degree orderings is finite; (3) the number of leading monomial ideals with respect to admissible orderings is finite under …
In this paper, we focus on the initial degree and the vanishing of the Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of $J$. For edge ideals of graphs, a complete characterization of the vanishing of the Valabrega-Valla module is given. For higher degree ideals, we find classes which the Valabrega-Valla module vanishes. For the case that $J$ is the facet ideal of a clutter $\mathcal{C}$ and $I$ is the defining idea…
In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also give a formula…
Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute the resurgence of ${\mathfrak p}$, the Waldschmidt constant of ${\mathfrak p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of ${\mathfrak p}$.
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.
Minimal Involutive Bases
本文提出构造极小对合多项式基的算法，定义两种新对合除法并证明其性质，确保算法正确终止，且极小对合基在给定序下唯一，可作典范基。
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…
We give explicit generators for ideals of two classes of subspace arrangements embedded in certain reflection arrangements, generalizing results of Li-Li and Kleitman-Lovasz. We also give minimal generators for the ideals of arrangements that arise in a natural way from the p-skeleton of an n-dimensional cube and discuss conditions under which the generators that we give form a Groebner basis.
Christophersen's problem for monomial algebras
本文在单项式代数类中解决了Christophersen问题，利用单项式理想的不可约分解组合结构证明了自同构群维数下界并刻画了等号情形，还给出了可约与不可约情形下的更强下界及达到者。
In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.
In this paper a new classification of monomial curves in $\mathbb{A}^4(\mathbbmss{k})$ is given. Our classification relies on the detection of those binomials and monomials that have to appear in every system of binomial generators of the defining ideal of the monomial curve; these special binomials and monomials are called indispensable in the literature. This way to proceed has the advantage of producing a natural necessary and sufficient condition for the definining ideal of a monomial curve in $\mathbb{A}^4(\mathbbmss{k})$ to have a unique minimal system of binomial generators. Furthermore…
In this paper, we consider a monomial ideal J in P := A[x1,...,xn], over a commutative ring A, and we face the problem of the characterization for the family Mf(J) of all homogeneous ideals I in P such that the A-module P/I is free with basis given by the set of terms in the Groebner escalier N(J) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involut…
In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gröbner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a g…
Let $K$ be an arbitrary field. Let $\a = (a_1< ... <a_n)$ be a sequence of positive integers. Let $C(\a)$ be the affine monomial curve in ${\mathbb A}^n$ parametrized by $t\to (t^{a_1}, ..., t^{a_n})$. Let $I(\a)$ be the defining ideal of $C(\a)$ in $K[x_1, ..., x_n]$. For each positive integer $j$, let $\a+j$ be the sequence $(a_1 + j, ..., a_n+j)$. In this paper, we prove the conjecture of Herzog and Srinivasan saying that the betti numbers of $I(\a + j)$ are eventually periodic in $j$ with period $a_n -a_1$. When $j$ is large enough, we describe the betti table for the closure of $C(\a+j)$ …
For polynomial ideals in positive charachteristic, defining $F$-split rings and admitting a squarefree monomial initial ideal are different notions. In this note we show that, however, there are strong interactions in both directions. Moreover we provide an overview on which $F$-singularities are Gröbner deforming. Also, we prove the following characteristic-free statement: if $\mathfrak{p}$ is a height $h$ prime ideal such that $\textrm{in}(\mathfrak{p}^{(h)})$ contains at least one squarefree monomial, then $\textrm{in}(\mathfrak{p})$ is a squarefree monomial ideal.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
We show that the defining ideal of every monomial curve in the affine or projective three-dimensional space can be set-theoretically defined by three binomial equations, two of which set-theoretically define a determinantal ideal generated by the 2-minors of a $2\times 3$ matrix with monomial entries.
Betti numbers of the tangent cones of monomial space curves
本文通过刻画数值半群环切锥定义理想的方程，证明了单项式空间曲线上切锥 Betti 数不超过其区间完备化对应 Betti 数的 Herzog-Stamate 猜想。
Extremal Betti numbers of certain two-dimensional monomial ideals
本文研究了多项式环上特定加权超平面定义的二维单项式理想，给出了 $R/I$ 极值 Betti 数的显式公式，并据此完全分类了此类伪 Gorenstein 理想。
On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
In this paper, we study a family of binomial ideals defining monomial curves in the $n-$dimensional affine space determined by $n$ hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with $u_{ii} = 0$, $i\in \{ 1, \ldots, n\}$. We prove that, the monomial curves in that family are set-theoretic complete intersection. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Fröbenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semig…
We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve. We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm. We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra.
Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves
本文研究 4 维空间中 Gorenstein 非完全交单项式曲线的切锥 Cohen-Macaulay 性，利用定义理想的极小生成元集给出判定准则，并刻画了相应理想极小生成元个数的可能取值。
Symbolic Rees algebras of space monomial primes of degree 5
本文研究了空间单项素理想度数5的符号Rees代数，证明其不是Noetherian的。
Let $d \geq 2$ and $m\geq 1$ be integers such that $\gcd (d,m)=1.$ Let ${\mathfrak p}$ be the defining ideal of the monomial curve in ${\mathbb A}_{ \Bbbk{k}}^d$ parametrized by $(t^{n_1}, \ldots, t^{n_d})$ where $n_i = d + (i-1)m$ for all $i = 1, \ldots, d$. In this paper, we describe the symbolic powers ${\mathfrak p}^{(n)} $ for all $n \geq 1$. As a consequence we show that the symbolic blowup algebras ${\mathcal R}_s{({\mathfrak p})}$ and $G_{s}({\mathfrak p}) $ are Cohen-Macaulay. This gives a positive answer to a question posed by S.~Goto in \cite{goto}. We also discuss when these blowup…
We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1]. We introduce switchable sets to give a combinatorial description of the minimal prime ideals, and for some classes we describe the minimal components. We discuss many possible interpretations of the ideals we study, including as 2 \times 2 minors of generic hypermatrices. We also introduce a definition of diagonal monomial orders on generic hypermatrices and…
We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.
We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves.
This article will appear in the proceedings of the AMS Summer Institute in Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by which I mean the defining ideals of subvarieties of affine or projective space which are parametrized by monomials. Numerous open problems are given.
Projective monomial curves associated to numerical semigroups with multiplicity $e$, width $e-1$, and embedding dimension $e-2$
本文研究重数为e、宽度为e-1、嵌入维数为e-2的数值半群S(e,m,n)所定义的射影单项曲线，得到其定义理想的Gröbner基，并刻画了曲线的Cohen–Macaulay与Gorenstein性质及正则度。
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