Abed Abedelfatah
We prove constructively that for any finite-dimensional commu- tative ring R, every stably free module over R[X;X^{1}] of rank > dim R is free, i.e., R[X;X^{-1}] is (dimR)-Hermite.
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If $R$ is an integral domain and $A$ is an $R$-algebra, then $A$ has the {\it Laurent cancellation property over $R$} if $A^{[\pm n]}\cong_RB^{[\pm n]}$ implies $A\cong_RB$ ($n\ge 0$ and $B$ an $R$-algebra). Here, $A^{[\pm n]}$ denotes the ring of Laurent polynomials in $n$ variables over $A$. Our main result (Thm. 4.3) is that, if the transcendence degree of $A$ over $R$ is one, then $A$ has the Laurent cancellation property. The proof uses the characterization of Laurent polynomial rings given in Thm. 3.2.
Factoriality and Class Groups of Upper Cluster Algebras and Finite Laurent Intersection Rings: A Computational Approach
本文提出了一种计算方法，用于研究上聚类代数的阶式和类群，通过引入有限拉格朗日交环并开发相关算法。
In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly closed field of characteristic 0. We define the multigrading of $S$ by an arbitrary finitely generated abelian group $A$. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of $S$ using polytopes. We give an algorithm to check whether the graded components of $S$ are finite dimensional. Regardless of …
We study the question if projective modules over formal Laurent series rings are extended. We relate this question to the Bass-Quillen conjecture for commutative regular local rings and to the Hermite ring conjecture for all commutative local rings. Using our result about projective modules over formal Laurent series rings, we prove a reduction step for the Hermite ring conjecture. We show that the Hermite ring conjecture holds for all commutative local rings if and only if it holds for complete intersection rings which are also unique factorization domains.
In this paper we proved that if $R$ is right $ω$-Euclidean domain, then skew Laurent formal series ring is right $ω$-Euclidean domain. We also showed that if $R$ is a right $ω$-Euclidean domain with multiplicative norm, then skew Laurent formal series ring is a right principal ideal domain. In addition, we proved that if $R$ is a noncommutative $ω$-Euclidean domain with a multiplicative norm, then $R$ and skew Laurent formal series ring is a ring with elementary reduction of matrices.
Pairings on the algebra of Laurent series over a ring
本文证明任意含单位交换环上洛朗级数环的连续不变双线性配对，在相差常数下等同于微分形式留数配对。
Dolgachev proved that, for any field k, the ring naturally associated to a generic Laurent polynomial in d variables, $d \geq 4$, is factorial. We prove a sufficient condition for the ring associated to a very general complex Laurent polynomial in d=3 variables to be Q-factorial.
A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of the order of the module. This description has applications in knot theory.
We first present a filtration on the ring L of Laurent polynomials such that the direct sum decomposition of its associated graded ring gr L agrees with the direct sum decomposition of gr L, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in gr L, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).
Power-closed ideals of polynomial and Laurent polynomial rings
本文刻画复多项式环与 Laurent 多项式环中的幂闭理想，研究相应闭包、内部算子，并完整描述主幂闭理想及一般幂闭理想的根。
We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that "up to the action of the group", these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic sta…
Let R be a Noetherian commutative ring of dimension n >2 and let A=R[T,T^{-1}]. Assume that the height of the Jacobson radical of R is atleast 2. Let P be a projective A-module of rank n=dim A - 1 with trivial determinant. We define an abelian group called the "Euler class group of A", denoted by E(A). Let χbe an isomorphism from A to det(P). To the pair (P,χ), we associate an element of E(A), called the Euler class of P, denoted by e(P,χ). Then we prove that a necessary and sufficient condition for P to have a unimodular element is the vanishing of e(P,χ) in E(A). Earlier, Bhatwadekar and Raj…
We say that a polynomial automorphism $ϕ$ in $n$ variables is stably co-tame if the tame subgroup in $n$ variables is contained in the subgroup generated by $ϕ$ and affine automorphisms in $n+1$ variables. In this paper, we give conditions for stably co-tameness of polynomial automorphisms.
Retracts of Laurent polynomial rings
本文研究多项式环对单项式局部化后的收缩核结构，证明当单项式为所有变量乘积时收缩核必为Laurent多项式环，并在完全域三维情形下给出收缩核的完整同构分类。
Construction of Bases in Modules over Laurent Polynomial Rings and Applications to Box Spline Prewavelets
本研究提出Laurent多项式模上线性形式核的基构造新方法，将其应用于箱样条预小波，在二元样条中实现更小掩模支撑，还构造出三元低非零系数分段线性预小波，优于现有构造。
An explicit non-elementary matrix over a bivariate Laurent polynomial ring
本文针对离散赋值环的分式域F，构造了二元Laurent多项式环上SL₂群中不属于初等子群E₂的显式矩阵，其非初等性证明依赖于特定预印本的技术。
If $i:A\subset B$ is a commutative ring extension, we show that the group $\mathcal I(A,B)$ of invertible $A$-submodules of $B$ is contracted in the sense of Bass, with $L\mathcal I(A,B)=H^0_{et}(A,i_*\mathbb Z/\mathbb Z)$. This gives a canonical decomposition for $\mathcal I(A[t,\frac1t],B[t,\frac1t])$.
Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Koh…
Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper we define the semistar dimension (inequality) formula and discover their relations with $\star$-universally catenarian domains and $\star$-stably strong S-domains. As an application we give new characterizations of $\star$-quasi-Prüfer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domains). We also extend Arnold's formula to the setting of semistar operations.
Let $R=\bigoplus_{n\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\dots,x_e$ of positive degrees $d_1,\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\bigoplus_{n\in\BZ}M_n$ the {\it Hilbert series\/} $\sum_{n\in\BZ}(\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function $$ H_M(t)=\frac{q_M(t)}{\prod_{i=1}^e(1-t^{d_i})} $$ with $q_M(t)\in\BZ[t,\ti]$. We demonstrate that Laurent expansions $\left[M\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\overline…
If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectivel…
Poincaré Duality and Quadratic Refinements over Laurent Rings
本文发展了Laurent环上配对缺陷的庞加莱对偶理论，通过球面三角剖分构造平坦分解，并利用等变上同调建立二次细化，进而证明高维泡利稳定子码辫子配对的非退化性。
In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.
Quasi-factorially closed subalgebras of Laurent polynomial rings
研究Laurent多项式环中准因子闭子代数的性质，通过局部化准则和群结构特征，揭示其与数值半群、代数闭性等概念的关系。
Finiteness and infiniteness of gradings of Noetherian rings
本文刻画了无挠阿贝尔群上存在非零分次的Noetherian环的充要条件为群秩有限，构造了相应的PID例子，并利用分次除环与群上同调证明了第二上同调的消失性及Hilbert级数的良定义性。
Let $A\subseteq B$ be a commutative ring extension. Let $\mathcal I(A, B)$ be the multiplicative group of invertible $A$-submodules of $B$. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension $A\subseteq B$ of integral domains with $\dim A\leq 1$, so that the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}],B [X, X^{-1}])$ is an isomorphism. In the same situation, we show that if $\dim A\geq 2$ then the condition is necessary but not sufficient. We also discuss some properties of the cokern…
This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to diophantine properties of the fields of Laurent series.
Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting of elements with bounded denominators. In previous joint work with Erman and Sam, we showed that $\mathcal{R}$ and $\mathcal{R}^{\flat}$ (and many similarly defined rings) are abstractly polynomial rings, and used this to give new proofs of Stillman's conjecture. In this paper, we prove the complementary result that $\mathcal{R}$ is a polynomial algebra ove…
Which series are Hilbert series of graded modules over polynomial rings?
该研究刻画了多重分次多项式环上有限生成模的Hilbert级数对应的形式Laurent级数，给出了给定深度模的Hilbert级数的必要条件，并完全分类了双分次下正深度有限生成模的Hilbert级数。
A counterexample to the Hermite ring conjecture
该研究构造出多项式环上非基环扩张的稳定自由模，给出埃尔米特环猜想的反例，推进了相关代数理论研究。
Linear Laurent phenomenon algebras are of finite type
本文在任意唯一分解整环上证明了Lam和Pylyavskyy提出的猜想，即具有线性种子的Laurent现象代数是有限型的，通过构造上代数的非零元素并结合初始Laurent环的素因子完成证明。
Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is \leq d. In case n=0, this result is due to Bhatwadekar, without the condition that f is a monic polynomial. (ii) The p-th Euler class group E^p(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.
A Nullstellensatz for the field of complex Laurent series
针对复Laurent级数域的理论，引入Kochen算子的-adic类似物及相关Kochen环，刻画可嵌入形式-adic域模型的域扩张，最终证明复Laurent级数域的零点定理，建立理想与代数集的对应关系并比较相关谱的不同形式。
Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and investigate the notions of $\star$-universally catenarian and $\star$-stably strong S-domains and prove that, every $\star$-locally finite dimensional Prüfer $\star$-multiplication domain is $\star$-universally catenarian, and this implies $\star$-stably strong S-domain. We also give new characterizations of $\star$-quasi-Prüfer domains introduced recently by Chang a…
In a recent paper [El 13], M.E. Kahoui has shown that if $R$ is a polynomial ring over $\mathbb{C}$, $A$ an $\mathbb{A}^3$-fibration over $R$, and $W$ a residual variable of $A$ then $A$ is stably polynomial over $R[W]$. In this article we show that the above result holds over any Noetherian domain $R$ provided the module of differentials $Ω_R(A)$ of the affine fibration $A$ (which is necessarily a projective $A$-module by a theorem of Asanuma) is a stably free $A$-module.
Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let $A=R_{g_1\cdots g_t} [Y_1,\ldots,Y_m,f_1(l_1)^{-1},\ldots, f_n(l_n)^{-1}]$, where $f_i(T)\in k[T]$ and $l_i=a_{i1}Y_1+\ldots+a_{im}Y_m$ with $(a_{i1},\ldots,a_{im})\in k^m-(0)$. Then every projective $A$-module of rank $\geq t$ is free. Laurent polynomial case $f_i(l_i)=Y_i$ of this result is due to Popescu.
From Ideal Membership Problem for polynomial rings to Dehn Functions of Metabelian Groups
本文定义整数系数Laurent多项式环中理想成员问题的复杂性函数，并将其与亚交换群的Dehn函数相联系，为构造超指数Dehn函数的亚交换群提供途径。
Let A be a commutative Noetherian ring of dimension d and let P be a projective R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then (i) \EL^1(R\op P) acts transitively on Um(R\oplus P). In particular, P is cancellative. (ii) If A is an affine algebra over a field, then P has a unimodular element. (iii) The natural map Φ_r : GL_r(R)/EL^1_r(R) \ra K_1(R) is surjective. (iv) Assume f_i is a monic polynomial. Then Φ_{r+1} is an isomorphism. In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel, (ii) is due…
Structure of modules stably annihilated by a fixed ideal
本文引入由稳定零化子包含固定理想I的有限生成模构成的子范畴，研究其结构，并将Dey-Liu定理从Gorenstein情形推广到Cohen-Macaulay情形。
Polynomial Extensions and Localization of Non-Noetherian Cohen--Macaulay Rings
研究非诺特环中Hamilton-Marley意义下Cohen-Macaulay（HMCM）性质的多项式扩张保持性，以及无挠模的局部化现象，给出了稳定凝聚环有限弱整体维数下的保持性，并刻画了局部化保持无挠性的条件。
The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely ge…
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple comple…
A cluster is a finite set of generators of a cluster algebra. The Laurent Phenomenon of Fomin and Zelevinsky says that any element of a cluster algebra can be written as a Laurent polynomial in terms of any cluster. The upper cluster algebra of a cluster algebra is the ring of rational functions that can be written as a Laurent polynomial in every cluster of the cluster algebra. By the Laurent phenomenon a cluster algebra is always contained in its upper cluster algebra, but they are not always equal. In 2014 it was conjectured that the equality of the cluster algebra and upper cluster algebra…
The approach of cluster symmetry to Diophantine equations
本文通过聚类对称性方法，提出了一种求解特定丢番图方程的新框架，并解决了多个相关数学问题。
Finite domination and Novikov homology over strongly $\mathbb{Z}^2$-graded rings
本文研究了强$\mathbb{Z}^2$-格rade环上链复形的有限支配性质，证明了在张量乘以特定的八重形式幂级数环后，复形成为单形的条件，扩展了Ranicki、Quinn和第一作者关于单变量和双变量 Laurent 多项式环的结果。
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