Laurent phenomenon sequences
In this paper, we undertake a systematic study of sequences generated by recurrences x[subscript m+n]x[subscript m]=P(x[subscript m+1],…,x[subscript m+n−1])xm+nxm=P(xm+1,…,xm+n−1) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences....
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Springer US
2016
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| Online Access: | http://hdl.handle.net/1721.1/103143 |
| _version_ | 1826191644514320384 |
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| author | Alman, Joshua Cuenca, Cesar Huang, Jiaoyang Cuenca, Cesar A. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Alman, Joshua Cuenca, Cesar Huang, Jiaoyang Cuenca, Cesar A. |
| author_sort | Alman, Joshua |
| collection | MIT |
| description | In this paper, we undertake a systematic study of sequences generated by recurrences x[subscript m+n]x[subscript m]=P(x[subscript m+1],…,x[subscript m+n−1])xm+nxm=P(xm+1,…,xm+n−1) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam–Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of n=2,3 and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver. |
| first_indexed | 2024-09-23T08:59:08Z |
| format | Article |
| id | mit-1721.1/103143 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:59:08Z |
| publishDate | 2016 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1031432022-09-30T12:38:58Z Laurent phenomenon sequences Alman, Joshua Cuenca, Cesar Huang, Jiaoyang Cuenca, Cesar A. Massachusetts Institute of Technology. Department of Mathematics Alman, Joshua Cuenca, Cesar A. Huang, Jiaoyang In this paper, we undertake a systematic study of sequences generated by recurrences x[subscript m+n]x[subscript m]=P(x[subscript m+1],…,x[subscript m+n−1])xm+nxm=P(xm+1,…,xm+n−1) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam–Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of n=2,3 and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver. National Science Foundation (U.S.) (grants DMS-1067183 and DMS-1148634) 2016-06-17T17:20:38Z 2017-03-01T16:14:48Z 2015-11 2014-09 2016-05-23T12:15:09Z Article http://purl.org/eprint/type/JournalArticle 0925-9899 1572-9192 http://hdl.handle.net/1721.1/103143 Alman, Joshua, Cesar Cuenca, and Jiaoyang Huang. “Laurent Phenomenon Sequences.” Journal of Algebraic Combinatorics 43, no. 3 (November 5, 2015): 589–633. en http://dx.doi.org/10.1007/s10801-015-0647-5 Journal of Algebraic Combinatorics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Alman, Joshua Cuenca, Cesar Huang, Jiaoyang Cuenca, Cesar A. Laurent phenomenon sequences |
| title | Laurent phenomenon sequences |
| title_full | Laurent phenomenon sequences |
| title_fullStr | Laurent phenomenon sequences |
| title_full_unstemmed | Laurent phenomenon sequences |
| title_short | Laurent phenomenon sequences |
| title_sort | laurent phenomenon sequences |
| url | http://hdl.handle.net/1721.1/103143 |
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