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A Recursive Method for Determining the One-Dimensional Submodules of Laurent-Ore Modules [article]

Ziming Li, Michael F. Singer, Min Wu, Dabin Zheng
2006 arXiv   pre-print
We present a method for determining the one-dimensional submodules of a Laurent-Ore module.  ...  The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules.  ...  In Section 5 we describe an algorithm for determining the one-dimensional submodules of an L-module and give some examples.  ... 
arXiv:cs/0604084v1 fatcat:4t4nrrftjre35kx2itklvmyzdy

A recursive method for determining the one-dimensional submodules of Laurent-Ore modules

Ziming Li, Michael F. Singer, Min Wu, Dabin Zheng
2006 Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06  
We present a method for determining the one-dimensional submodules of a Laurent-Ore module.  ...  The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules.  ...  In Section 5 we describe an algorithm for determining the one-dimensional submodules of an L-module and give some examples.  ... 
doi:10.1145/1145768.1145806 dblp:conf/issac/LiSWZ06 fatcat:zsiqmdripjao7oyn3gazup4z4a

ON SOLUTIONS OF LINEAR FUNCTIONAL SYSTEMS AND FACTORIZATION OF LAURENT–ORE MODULES

MIN WU, ZIMING LI
2007 Computer Algebra 2006  
A generalized Beke's method is also presented for factoring Laurent-Ore modules and it will allow us to find all "subsystems"whose solution spaces are contained in that of a given linear functional system  ...  By associating a finite-dimensional linear functional system to a Laurent-Ore module, Picard-Vessiot extensions are generalized from linear ordinary differential (difference) equations to finite-dimensional  ...  The first subproblem can be solved by a recursive method [15] for determining onedimensional submodules of a Laurent-Ore module.  ... 
doi:10.1142/9789812778857_0007 fatcat:yfepuxw3rndfnhx7w3tjfxicpi

Transforming linear functional systems into fully integrable systems

Ziming Li, Min Wu
2012 Journal of symbolic computation  
The algorithm avoids using Gröbner bases in Laurent-Ore modules when ∂-finite systems correspond to finitedimensional Ore modules.  ...  A linear (partial) functional system consists of linear partial differential, difference equations or any mixture thereof.  ...  Acknowledgements We thank Professor Michael Singer for encouraging us to describe the linear reduction in Wu (2005, §2.5.2) using module-theoretic language.  ... 
doi:10.1016/j.jsc.2011.12.028 fatcat:24js5ghpyjgyzj7eoa4no7ruf4

Novel representation of discrete n-D autonomous systems [article]

Debasattam Pal, Harish K. Pillai
2015 arXiv   pre-print
In this result we show that every quotient ring of the n-variable Laurent polynomial ring can be made a finitely generated faithful module over another Laurent polynomial ring of smaller dimension by doing  ...  We give a full description of the set of allowable initial conditions. In our search for a general representation formula, one algebraic result plays a very crucial role.  ...  This fact together with Swan's extension [Swa78] of Serre conjecture to Laurent polynomial rings give us the following check (Theorem 32) for determining whether A q /R admits a free X or not.  ... 
arXiv:1505.05684v1 fatcat:gavwvomu2zdc5jypkji5x62iou

On the quiver Grassmannian in the acyclic case

Philippe Caldero, Markus Reineke
2008 Journal of Pure and Applied Algebra  
Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M.  ...  Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.  ...  The second author wants to thank Klaus Bongartz for spotting a mistake in an earlier version of the paper.  ... 
doi:10.1016/j.jpaa.2008.03.025 fatcat:ezfmep7wprdvpkx4cw33en5dqa

Universal central extensions of elliptic affine Lie algebras

Murray Bremner
1994 Journal of Mathematical Physics  
The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined.  ...  Let g be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed.  ...  I thank David Roberts for discussions on the cohomology of algebraic curves, Rui-ming Zhang for Ref. 19, and the referee for a very helpful report.  ... 
doi:10.1063/1.530700 fatcat:d76qjtjnsnghnfsp6h6tpfi6ni

On rings of invariants of non-modular Abelian groups

H.E.A. Campbell, J.C. Harris, D.L. Wehlau
1999 Bulletin of the Australian Mathematical Society  
We study the ring of invariant Laurent polynomials associated to the action of a finite diagonal group G on the symmetric algebra of a vector space over a field F.  ...  In particular we obtain a recursive formula for the number of minimal generators for these rings of invariants.  ...  LAURENT WEIGHT SUBMODULES FOR CYCLIC GROUPS. We denote by K(w) the set of Laurent polynomials with basis the monomials of K of weight w for 0 $J w < q.  ... 
doi:10.1017/s0004972700036674 fatcat:rpncau5m5fdu7fgkefg4iqfhti

Locally free Caldero-Chapoton functions via reflections [article]

Lang Mou
2022 arXiv   pre-print
Our method gives rise to a new proof of the locally free Caldero-Chapoton formulas obtained by Geiss-Leclerc-Schr\"oer in Dynkin cases.  ...  We study the reflections of locally free Caldero-Chapoton functions associated to representations of Geiss-Leclerc-Schr\"oer's quivers with relations for symmetrizable Cartan matrices.  ...  The key recursion. The following is the key proposition on the recursion of F -polynomials under reflections. Proposition 4.7. Let M ∈ rep l.f. H be of rank (m i ) i∈I and k be a sink of H.  ... 
arXiv:2206.02289v1 fatcat:wvakioh7rzc4loo23k6fwikvxy

Friezes, Strings and Cluster Variables [article]

Ibrahim Assem, Grégoire Dupont, Ralf Schiffler, David Smith
2011 arXiv   pre-print
When the walk is the string of a string module over a 2-Calabi-Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module, up to an  ...  To any walk in a quiver, we associate a Laurent polynomial.  ...  Acknowledgments Work on this problem was started during the 2010 South American Meeting on Representations of Algebras and Related Topics in Mar del Plata (Argentina); the authors wish to thank the organisers  ... 
arXiv:1009.3341v2 fatcat:igtamq42qjh6vpfcy4vwushecy

On representations of U'_qso_n [article]

Hans Wenzl
2019 arXiv   pre-print
This is used to recover the classification of finite-dimensional modules for q not a root of unity, given by classical and non-classical series.  ...  We study representations of the non-standard quantum deformation U'_qso_n of Uso_n via a Verma module approach.  ...  In the third section, we prove all the necessary results for Verma modules for Uq′ so3 via elementary methods which more or less have been known before.  ... 
arXiv:1805.06268v2 fatcat:swyo6qliyzb27itjmn5niwhd4y

From triangulated categories to cluster algebras II [article]

Philippe Caldero, Bernhard Keller
2006 arXiv   pre-print
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra.  ...  This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph.  ...  Note that the morphism H 0 ζ ′ = ζ ′ is non zero. Now, the lemma implies that for any submodule M ′ of M , M ′ is a either submodule of im H 0 π or contains ker H 0 π ′ .  ... 
arXiv:math/0510251v2 fatcat:4lsmuc5rfvdedkzeudcrfymody

From triangulated categories to cluster algebras II

P CALDERO, B KELLER
2006 Annales Scientifiques de l'Ecole Normale Supérieure  
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra.  ...  This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph.  ...  The authors thank Andrew Hubery for pointing out a gap in a previous version of this article.  ... 
doi:10.1016/j.ansens.2006.09.003 fatcat:br6iyj6t2vcqdbu6oq5j24vnkm

FRIEZES, STRINGS AND CLUSTER VARIABLES

IBRAHIM ASSEM, GRÉGOIRE DUPONT, RALF SCHIFFLER, DAVID SMITH
2011 Glasgow Mathematical Journal  
When the walk is the string of a string module over a 2-Calabi-Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an  ...  We first obtain the positivity of the Laurent polynomial X T M for any string B T -module M (see Corollary 6.4), thus reproving the result of Cerulli and Haupt [18, 34] .  ...  In the previous section we associated with any string module M over a finite dimensional algebra B a certain Laurent polynomial L M .  ... 
doi:10.1017/s0017089511000322 fatcat:dy65ofwpvvf5lbzwne5bv6tche

Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations

Franz Pauer, Andreas Unterkircher
1999 Applicable Algebra in Engineering, Communication and Computing  
Motivation and Introduction Let R be a commutative noetherian ring (e. g. a field, 9 or 9 m ), a set, let R be the R-module of all maps from to R, and let R ( ) be the R-submodule of all maps from to R  ...  Furthermore, we present a method to compute the intersection of an ideal in the algebra of Laurent polynomials with the subalgebra of all polynomials.  ...  Examples Let F be a finite subset of V \ {0}. In order to compute a Gröbner basis of the submodule generated by F , we first have to determine the sets T i (f ), for all i ∈ I , f ∈ F .  ... 
doi:10.1007/s002000050108 fatcat:reuxkdwpdzh5fhefldgvyb4ql4
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