Bjarke Hammersholt Roune
Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.
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Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field $\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum number $k$ for which $I \colon \mathfrak{m}^k = I \colon \mathfrak{m}^{k+1}$. In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appea…
The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that the second algorithm is more efficient than the first one for generic monomial ideals. Furthermore, the time complexity of the second algorithm is at most $O(n^2p\ell)$ where $n$ is the number of variables…
Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an arbitrary commutative ring with identity. We classify the irreducible elements of this set, which we call m-irreducible, and we classify the elements that admit decompositions into finite intersections of m-irreducible ideals.
Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numeri…
We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to compute $v$ is given using Macaulay2 (Section 7). For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph (Proposition 4.3). For ideals of Borel type the monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when $f$ is unique (Proposition 6.2).
Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case the Cohen-Macaulay type is greater than or equal to the numbe…
Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n-1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions.
Waldschmidt constant of monomial ideals and Simis ideals
本文验证了某些单项式理想类别的猜想，表明若理想满足特定条件，则可表示为Simis自由单项式理想与标准线性加权的组合。
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。
Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree $i$ has dimension exactly dim(R/I) - i. This Cohen-Macaulay characterization reduces to the Eagon-Reiner theorem by Alexander duality when R is a polynomial ring. The proof exploits a graded ring-theoretic generalization of the Zeeman spectral sequence, thereby also providing a combinatorial topological version for polyhedral cel…
We compute the Stanley depth of irreducible monomial ideals and we show that the Stanley depth of a monomial complete intersection ideal is the same as the Stanley depth of it's radical. Also, we give some bounds for the Stanley depth of a monomial complete intersection ideal.
Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals
本文利用组合优化与整数规划研究覆盖多面体滤过的Rees代数，计算Waldschmidt常数与ic-回归，给出图覆盖理想和边理想的下界，并分类牛顿多面体为不可约多面体的条件。
Let $Q$ and $Q'$ be two monomial primary ideals of a polynomial algebra $S$ over a field. We give an upper bound for the Stanley depth of $S/(Q\cap Q')$ which is reached if $Q$,$Q'$ are irreducible. Also we show that Stanley's Conjecture holds for $Q_1\cap Q_2$, $S/(Q_1\cap Q_2\cap Q_3)$, $(Q_i)_i$ being some irreducible monomial ideals of $S$.
Support-2 monomial ideals that are Simis
本文研究了支持2的单项式理想在Simis性质下的条件，证明了其与radical的Simis性质及标准线性权重的关系，并探讨了Cohen-Macaulay性质及二次Simis性质的分类。
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equ…
The Mendez-Pinto-Villarreal Conjecture for some classes of monomial ideals
研究交换代数中刻画理想符号幂与普通幂重合问题（西米斯理想），通过研究门德斯 - 平托 - 比利亚雷亚尔猜想，利用支撑集条件证明相关结论，建立两类单项式理想的猜想并研究其科恩 - 麦考利性质。
Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.
Given a free resolution of an ideal $J$ of holomorphic functions, one can construct a vector valued residue current $R$, whose annihilator is precisely $J$. In this paper we compute $R$ in case $J$ is a monomial ideal and the resolution is a cellular resolution in the sense of Bayer and Sturmfels. A description of $R$ is given in terms of the underlying polyhedral cell complex and it is related to irreducible decompositions of $J$.
We present stdPairs.spyx, a SageMath library to compute standard pairs of a monomial ideal over a pointed (non-normal) affine semigroup ring. Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from arXiv:2005.10968v2 [math.AC]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.
$\operatorname{v}$-numbers of integral closure filtrations of monomial ideals
本文研究了多项式环中单项式理想幂及其整闭包的$\operatorname{v}$-数，分析了不可约和完全交单项式理想整闭包滤过的$\operatorname{v}$-数与Castelnuovo-Mumford正则性的关系，并给出了相关公式与上界。
Signature invariants of monomial ideals
研究单项理想特征量的性质，证明其深度、正则度及Cohen-Macaulay/Gorenstein性质的等价性，并提供计算特征理想的方法。
Let $I\subset J$ be monomial ideals of a polynomial algebra $S$ over a field. Then the Stanley depth of $J/I$ is smaller or equal with the Stanley depth of $\sqrt{J}/\sqrt{I}$. We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals $Q$, $Q'$, which is reached if $Q$, $Q'$ are irreducible, ht$(Q+Q')$ is odd and $\sqrt{Q}$, $\sqrt{Q'}$ have no common variable.
Associated primes, witnesses, and omega invariants of monomial ideals
本文引入诺特环中理想的omega不变量（相伴素理想个数），通过单项式见证元与矩阵方法刻画单项式理想及其幂的相伴素，给出无需准素分解的公式与界，并推广至边理想。
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…
Infinite Minkowski sums of lattice polyhedra
本文研究一类可唯一表示为整闭单项式理想无限乘积的单项式子模，其牛顿多面体对应简单格多面体的无限闵可夫斯基和，从而构造出拓扑唯一分解整环的新例子。
The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals
本文研究高度为 h 的单项式理想，在其无冗余约化准素分解中高度 h 准素理想均不可约的条件下，推广已有定理，给出理想幂、特殊幂的幂，以及加权有向图边理想相应情形的重数公式。
On unboundedness of some invariants of $\mathcal{C}$-semigroups
本文研究了C-半群的类型和减少类型在嵌入维数下的无界性，并给出了不可约分解的下界。
In this note we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [8, 5] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes i…
Essential graded algebra over polynomial rings with real exponents
本文在实指数多项式环及实多面体锥的幺半群代数上建立单项式理想与多重分次模的分解理论，推广中山引理，并证明半代数与分段线性模的极小分解具有典范性。
The purpose of this note is to find an elemenary explanation of a surprising result of Ein--Lazarsfeld--Smith \cite{ELS} and Hochster--Huneke \cite{HH} on the containment between symbolic and ordinary powers of ideals in simple cases. This line of research has been very active ever since, see for instance \cites{BC,HaH,DST} and the references therein, by now the literature on this topic is quite extensive. By `elementary' we refer to arguments that among others do not make use of resolution of singularities and multiplier ideals nor tight closure methods. Let us quickly recall the statement \c…
The Gröbner stratum of a monomial ideal $\id{j}$ is an affine variety that parametrizes the family of all ideals having $\id{j}$ as initial ideal (with respect to a fixed term ordering). The Gröbner strata can be equipped in a natural way of a structure of homogeneous variety and are in a close connection with Hilbert schemes of subvarieties in the projective space $\PP^n$. Using properties of the Gröbner strata we prove some sufficient conditions for the rationality of components of $\hilb_{p(z)}^n$. We show for instance that all the smooth, irreducible components in $\hilb_{p(z)}^n$ (or in i…
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