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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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Інститут математики НАН України
2012
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| Zitieren: | 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. |
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Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. 2019-02-18T17:38:16Z 2019-02-18T17:38:16Z 2012 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 https://nasplib.isofts.kiev.ua/handle/123456789/148658 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. This paper is a contribution to the Special Issue “Mirror Symmetry and Related Topics”. The full collection is available at http://www.emis.de/journals/SIGMA/mirror symmetry.html. We thank Alessio Corti and Vasily Golyshev for many useful conversations, the referees for perceptive and helpful comments, John Cannon for providing copies of the computer algebra software Magma, and Andy Thomas for technical assistance. This research is supported by a Royal Society University Research Fellowship; ERC Starting Investigator Grant number 240123; the Leverhulme Trust; Kavli Institute for the Physics and Mathematics of the Universe (WPI); World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan; Grant-in-Aid for Scientific Research (10554503) from Japan Society for Promotion of Science and Grant of Leading Scientific Schools (N.Sh. 4713.2010.1); and EPSRC grant EP/I008128/1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Minkowski Polynomials and Mutations Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Minkowski Polynomials and Mutations |
| spellingShingle |
Minkowski Polynomials and Mutations Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. |
| 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 |
| author |
Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. |
| author_facet |
Akhtar, M. Coates, T. Galkin, S. Kasprzyk, A.M. |
| publishDate |
2012 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.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. |
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2025-12-07T20:16:26Z |
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2025-12-07T20:16:26Z |
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