scholarly journals Positivity in Coefficient-Free Rank Two Cluster Algebras

10.37236/187 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
G. Dupont
Keyword(s):  
Short Note ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Free Rank ◽  
Positive Integers

Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over ${\Bbb Z}$ and $x_m, m \in {\Bbb Z}$ be rational functions defined by $x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k \in {\Bbb Z}$, $x_k$ can be expressed as a substraction-free Laurent polynomial in ${\Bbb Z}[x_m^{\pm 1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.

scholarly journals Links in the complex of weakly separated collections

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Suho Oh ◽  
David Speyer
Keyword(s):  
Short Note ◽  
Cluster Algebras ◽  
Combinatorial Objects ◽  
The Face ◽  
International Audience ◽  
Plabic Graphs ◽  
Positive Integers ◽  
Do So

International audience Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals The simultaneous representation of integers by products of certain rational functions

1983 ◽  
Vol 35 (3) ◽  
pp. 405-420
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Rational Functions ◽  
Linear Recurrences ◽  
Integer Points ◽  
Power Sums ◽  
Representation Of Integers ◽  
Simultaneous Representation ◽  
Positive Integers ◽  
By Products

AbstractIt is proved that an arbitrary pair of positive integers can be simultaneously represented by products of the values at integer points of certain rational functions. Linear recurrences in Z-modules and elliptic power sums are applied.

scholarly journals Cluster algebras of unpunctured surfaces and snake graphs

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gregg Musiker ◽  
Ralf Schiffler
Keyword(s):  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Polynomial Expansion ◽  
International Audience ◽  
Polynomial Expansions

International audience We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ . Nous étudions des algèbres amassées avec coefficients principaux associées aux surfaces. Nous présentons une formule directe pour les développements de Laurent des variables amassées dans ces algèbres en terme de couplages parfaits d'un certain graphe $G_{T,\gamma}$ que l'on construit a partir de la surface en recollant des pièces élémentaires que l'on appelle carreaux. Nous donnons aussi une seconde formule pour ces développements en termes de sous-graphes de $G_{T,\gamma}$ .

Restrictions on meromorphic solutions of Fermat type equations

2020 ◽  
Vol 63 (3) ◽  
pp. 654-665
Author(s):  
Gary G. Gundersen ◽  
Katsuya Ishizaki ◽  
Naofumi Kimura
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Existence Theorems ◽  
Rational Functions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Nevanlinna Functions ◽  
Positive Integers

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors

2014 ◽  
Vol 57 (3) ◽  
pp. 562-572
Author(s):  
Kiumars Kaveh ◽  
A. G. Khovanskii
Keyword(s):  
Algebraic Variety ◽  
Short Note ◽  
Grothendieck Group ◽  
Rational Functions ◽  
Intersection Theory ◽  
Point Of View ◽  
Intersection Numbers ◽  
Ground Field

Abstract.In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety X over k = ℂ. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field k. Secondly we give an isomorphism between the group of Cartier b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier b-divisors and their intersection theory.

Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field

2008 ◽  
Vol 60 (4) ◽  
pp. 923-957 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Height Function ◽  
Rational Functions ◽  
Trivial Extension ◽  
Closed Field ◽  
Quadratic Algebra ◽  
Endomorphism Algebra ◽  
Finite Dimensional ◽  
Free Rank ◽  
Rank 2 ◽  
Coordinate Rings

AbstractThe Kronecker modules , where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X over an algebraically closed field K, aremodels for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial f in K(X)[Y]. When the endomorphism algebra of is commutative and non-trivial, the regulator f must be quadratic in Y. If f has one repeated root in K(X), the endomorphismalgebra is the trivial extension for some vector space S. If f has distinct roots in K(X), then the endomorphisms forma structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End that are domains have zero radical. In addition, each semi-local End must be either a trivial extension or the product K × K.

A Leopoldt-Type Result for Rings of Integers of Cyclotomic Extensions

1995 ◽  
Vol 38 (2) ◽  
pp. 141-148 ◽  
Author(s):  
W. Bley
Keyword(s):  
Prime Number ◽  
Group Ring ◽  
Maximal Order ◽  
Type Result ◽  
Rings Of Integers ◽  
Cyclotomic Extensions ◽  
Rank One ◽  
Free Rank ◽  
Positive Integers

AbstractLet p be a prime number and let m, r denote positive integers with r ≥ 1 if p > 3 (resp. r ≥ 2 if p = 2) and m ≥ 1. We put and Γ = Gd1(N/M). Then the associated order of N/M is the unique maximal order M in the group ring MΓ and ON is a free, rank one module over M. A generator of ON over M is explicitly given.

scholarly journals Waring’s Problem for Rational Functions in One Variable

2020 ◽  
Vol 71 (2) ◽  
pp. 439-449
Author(s):  
Bo-Hae Im ◽  
Michael Larsen
Keyword(s):  
Rational Function ◽  
Sufficient Conditions ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Odd Degree ◽  
Necessary And Sufficient ◽  
Polynomial Of Odd Degree

Abstract Let $f\in{\mathbb{Q}}(x)$ be a non-constant rational function. We consider ‘Waring’s problem for $f(x)$,’ i.e., whether every element of ${\mathbb{Q}}$ can be written as a bounded sum of elements of $\{f(a)\mid a\in{\mathbb{Q}}\}$. For rational functions of degree $2$, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the ‘easier Waring’s problem’: whether every element of ${\mathbb{Q}}$ can be represented as a bounded sum of elements of $\{\pm f(a)\mid a\in{\mathbb{Q}}\}$.

scholarly journals Friezes and continuant polynomials with parameters

2018 ◽  
Vol 36 (2) ◽  
pp. 57-81
Author(s):  
Véronique Bazier-Matte ◽  
David Racicot-Desloges ◽  
Tanna Sánchez McMillan
Keyword(s):  
Finite Type ◽  
Type A ◽  
Cluster Algebras ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Family ◽  
Laurent Phenomenon ◽  
Necessary And Sufficient ◽  
Positive Integers

Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c-friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c-friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c-friezes of integers and of positive integers. We also show some specific properties of c-friezes.

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