Geodesic
Positivity in coefficient-free rank two cluster algebras.
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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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.
DOI : 10.37236/187
Classification : 16G20, 13F60, 05E10
Mots-clés : rational functions, substraction-free Laurent polynomials, Fomin-Zelevinsky positivity conjecture, rank two cluster algebras 
@article{10_37236_187,
     author = {G. Dupont},
     title = {Positivity in coefficient-free rank two cluster algebras.},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/187},
     zbl = {1193.16017},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/187/}
}
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G. Dupont. Positivity in coefficient-free rank two cluster algebras.. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/187

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