Minkowski Polynomials and Mutations
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the...
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| Дата: | 2012 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2012
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148658 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Minkowski Polynomials and Mutations / M. Akhtar, T. Coates, S. Galkin, A.M. Kasprzyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. |
Репозитарії
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irk-123456789-1486582019-02-19T01:28:03Z Minkowski Polynomials and Mutations Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period. 2012 Article Minkowski Polynomials and Mutations / M. Akhtar, T. Coates, S. Galkin, A.M. Kasprzyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 52B20; 16S34; 14J33 DOI: http://dx.doi.org/10.3842/SIGMA.2012.094 http://dspace.nbuv.gov.ua/handle/123456789/148658 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period. |
| format |
Article |
| author |
Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. |
| spellingShingle |
Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. Minkowski Polynomials and Mutations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. |
| author_sort |
Akhtar, M. |
| title |
Minkowski Polynomials and Mutations |
| title_short |
Minkowski Polynomials and Mutations |
| title_full |
Minkowski Polynomials and Mutations |
| title_fullStr |
Minkowski Polynomials and Mutations |
| title_full_unstemmed |
Minkowski Polynomials and Mutations |
| title_sort |
minkowski polynomials and mutations |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148658 |
| citation_txt |
Minkowski Polynomials and Mutations / M. Akhtar, T. Coates, S. Galkin, A.M. Kasprzyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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