## Period integrals and mutation

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## Abstract:

We define what it means for a Laurent polynomial in two variables to be*mutable*. For a mutable Laurent polynomial we prove several results about $f$ and its period $\pi _f$ in terms of the Newton polygon of $f$. In particular, we give an in principle complete description of the monodromy of $\pi _f$ around the origin. Special attention is given to the class of

*maximally mutable*Laurent polynomials, which has applications to the conjectured classification of Fano manifolds via mirror symmetry.

## References

- Mohammad Akhtar, Tom Coates, Alessio Corti, Liana Heuberger, Alexander Kasprzyk, Alessandro Oneto, Andrea Petracci, Thomas Prince, and Ketil Tveiten,
*Mirror symmetry and the classification of orbifold del Pezzo surfaces*, Proc. Amer. Math. Soc.**144**(2016), no. 2, 513–527. MR**3430830**, DOI 10.1090/proc/12876 - Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexander M. Kasprzyk,
*Minkowski polynomials and mutations*, SIGMA Symmetry Integrability Geom. Methods Appl.**8**(2012), Paper 094, 17. MR**3007265**, DOI 10.3842/SIGMA.2012.094 - Mohammad Akhtar and Alexander Kasprzyk,
*Singularity content*, arXiv:1401.5458v1 (2014). - A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers,
*Algebraic $D$-modules*, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR**882000** - Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander Kasprzyk,
*Mirror symmetry and Fano manifolds*, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, pp. 285–300. MR**3469127** - David A. Cox, John B. Little, and Henry K. Schenck,
*Toric varieties*, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR**2810322**, DOI 10.1090/gsm/124 - Alexandru Dimca,
*Sheaves in topology*, Universitext, Springer-Verlag, Berlin, 2004. MR**2050072**, DOI 10.1007/978-3-642-18868-8 - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR**1288523**, DOI 10.1002/9781118032527 - Mark Gross, Paul Hacking, and Sean Keel,
*Birational geometry of cluster algebras*, Algebr. Geom.**2**(2015), no. 2, 137–175. MR**3350154**, DOI 10.14231/AG-2015-007 - Sergey Galkin and Alexandr Usnich,
*Mutations of potentials*, preprint IPMU 10-0100 (2010). - Lars Hörmander,
*The analysis of linear partial differential operators. I*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR**1065993**, DOI 10.1007/978-3-642-61497-2 - Alexander M. Kasprzyk and Benjamin Nill,
*Fano polytopes*, Strings, Gauge Fields, and the Geometry Behind – the Legacy of Maximilian Kreuzer (Anton Rebhan, Ludmil Katzarkov, Johanna Knapp, Radoslav Rashkov, and Emanuel Scheidegger, eds.), World Scientific, 2012, pp. 349–364. - Alexander M. Kasprzyk and Ketil Tveiten,
*Maximally mutable laurent polynomials*, in preparation (2016). - Pierre Lairez,
*Computing periods of rational integrals*, Math. Comp.**85**(2016), no. 300, 1719–1752. MR**3471105**, DOI 10.1090/mcom/3054 - Alessandro Oneto and Andrea Petracci,
*On the quantum periods of del pezzo surfaces with $\frac 13(1,1)$ singularities*, in preparation (2015). - Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama,
*Gröbner deformations of hypergeometric differential equations*, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR**1734566**, DOI 10.1007/978-3-662-04112-3 - Henryk Żołądek,
*The monodromy group*, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 67, Birkhäuser Verlag, Basel, 2006. MR**2216496**

## Additional Information

**Ketil Tveiten**- Affiliation: Department of Mathematics,Uppsala University, Box 256,75105 Uppsala,Sweden
- MR Author ID: 1113057
- Email: ketiltveiten@gmail.com
- Received by editor(s): March 3, 2015
- Received by editor(s) in revised form: October 13, 2015, April 22, 2016, and March 17, 2017
- Published electronically: July 5, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8377-8401 - MSC (2010): Primary 32S40, 14J33
- DOI: https://doi.org/10.1090/tran/7320
- MathSciNet review: 3864380