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

2006
*
arXiv
*
pre-print

We present

arXiv:cs/0604084v1
fatcat:4t4nrrftjre35kx2itklvmyzdy
*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. ...##
###
A recursive method for determining the one-dimensional submodules of Laurent-Ore modules

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

We present

doi:10.1145/1145768.1145806
dblp:conf/issac/LiSWZ06
fatcat:zsiqmdripjao7oyn3gazup4z4a
*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. ...##
###
ON SOLUTIONS OF LINEAR FUNCTIONAL SYSTEMS AND FACTORIZATION OF LAURENT–ORE MODULES

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*. ...

##
###
Transforming linear functional systems into fully integrable systems

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. ...

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

2015
*
arXiv
*
pre-print

In this result we show that every quotient ring

arXiv:1505.05684v1
fatcat:gavwvomu2zdc5jypkji5x62iou
*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. ...##
###
On the quiver Grassmannian in the acyclic case

2008
*
Journal of Pure and Applied Algebra
*

Let

doi:10.1016/j.jpaa.2008.03.025
fatcat:ezfmep7wprdvpkx4cw33en5dqa
*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. ...##
###
Universal central extensions of elliptic affine Lie algebras

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. ...

##
###
On rings of invariants of non-modular Abelian groups

1999
*
Bulletin of the Australian Mathematical Society
*

We study

doi:10.1017/s0004972700036674
fatcat:rpncau5m5fdu7fgkefg4iqfhti
*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. ...##
###
Locally free Caldero-Chapoton functions via reflections
[article]

2022
*
arXiv
*
pre-print

Our

arXiv:2206.02289v1
fatcat:wvakioh7rzc4loo23k6fwikvxy
*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. ...##
###
Friezes, Strings and Cluster Variables
[article]

2011
*
arXiv
*
pre-print

When

arXiv:1009.3341v2
fatcat:igtamq42qjh6vpfcy4vwushecy
*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 ...##
###
On representations of U'_qso_n
[article]

2019
*
arXiv
*
pre-print

This is used to recover

arXiv:1805.06268v2
fatcat:swyo6qliyzb27itjmn5niwhd4y
*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. ...##
###
From triangulated categories to cluster algebras II
[article]

2006
*
arXiv
*
pre-print

In

arXiv:math/0510251v2
fatcat:4lsmuc5rfvdedkzeudcrfymody
*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 ′ . ...##
###
From triangulated categories to cluster algebras II

2006
*
Annales Scientifiques de l'Ecole Normale Supérieure
*

In

doi:10.1016/j.ansens.2006.09.003
fatcat:br6iyj6t2vcqdbu6oq5j24vnkm
*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. ...##
###
FRIEZES, STRINGS AND CLUSTER VARIABLES

2011
*
Glasgow Mathematical Journal
*

When

doi:10.1017/s0017089511000322
fatcat:dy65ofwpvvf5lbzwne5bv6tche
*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 . ...##
###
Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations

1999
*
Applicable Algebra in Engineering, Communication and Computing
*

Motivation and Introduction Let R be

doi:10.1007/s002000050108
fatcat:reuxkdwpdzh5fhefldgvyb4ql4
*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 . ...
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