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The maximum likelihood degree of Fermat hypersurfaces

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Abstract (2. Language): 
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the di erent methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
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REFERENCES

References: 

1] J.-E. Bjork. Rings of dierential operators, volume 21 of North-Holland Mathematical
Library. North-Holland Publishing Co., Amsterdam, 1979.
[2] Max-Louis G. Buot, Serkan Hosten, and Donald St. P. Richards. Counting and
locating the solutions of polynomial systems of maximum likelihood equations. II.
The Behrens-Fisher problem. Statist. Sinica, 17(4):1343{1354, 2007.
[3] Fabrizio Catanese, Serkan Hosten, Amit Khetan, and Bernd Sturmfels. The maximum
likelihood degree. Amer. J. Math., 128(3):671{697, 2006.
[4] Jan Draisma and Jose Rodriguez. Maximum likelihood duality for determinantal
varieties. Int. Math. Res. Not. IMRN, 20:5648{5666, 2014.
[5] Mathias Drton, Bernd Sturmfels, and Seth Sullivant. Lectures on algebraic statistics,
volume 39 of Oberwolfach Seminars. Birkhauser Verlag, Basel, 2009.
[6] David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 1995. With a view toward algebraic geometry.
[7] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research
in algebraic geometry. available at www.math.uiuc.edu/Macaulay2/.
[8] Elizabeth Gross, Mathias Drton, and Sonja Petrovic. Maximum likelihood degree of
variance component models. Electron. J. Stat., 6:993{1016, 2012.
[9] Elizabeth Gross and Jose Rodriguez. Maximum likelihood geometry in the presence
of data zeros. In ISSAC 2014 | Proceedings of the 39th International Symposium on
Symbolic and Algebraic Computation, 2014.
REFERENCES 132
[10] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977.
[11] Jonathan Hauenstein, Jose Rodriguez, and Bernd Sturmfels. Maximum likelihood for
matrices with rank constraints. J. Alg. Stat., 5(1):18{38, 2014.
[12] Serkan Hosten, Amit Khetan, and Bernd Sturmfels. Solving the likelihood equations.
Found. Comput. Math., 5(4):389{407, 2005.
[13] Serkan Hosten and Seth Sullivant. The algebraic complexity of maximum likelihood
estimation for bivariate missing data. In Algebraic and geometric methods in statistics,
pages 123{133. Cambridge Univ. Press, Cambridge, 2010.
[14] June Huh. The maximum likelihood degree of a very ane variety. Compos. Math.,
149(8):1245{1266, 2013.
[15] June Huh. Varieties with maximum likelihood degree one. J. Alg. Stat., 5(1):1{17,
2014.
[16] June Huh and Bernd Sturmfels. Likelihood geometry. In Sandra Di Rocco and Bernd
Sturmfels, editors, Combinatorial Algebraic Geometry, volume 2108 of Lecture Notes
in Mathematics (C.I.M.E. Foundation Subseries). Springer, 2014.
[17] Joachim Jelisiejew. Deformations of zero-dimensional schemes and applications. Master's
thesis, University of Warsaw, 2013. Available at arxiv.org/abs/1307.8108.
[18] Ezra Miller and Bernd Sturmfels. Combinatorial commutative algebra, volume 227 of
Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.
[19] Lior Pachter and Bernd Sturmfels. Algebraic Statistics for Computational Biology.
Cambridge Univ. Press, 2005.
[20] Caroline Uhler. Geometry of maximum likelihood estimation in Gaussian graphical
models. Ann. Statist., 40(1):238{261, 2012.
[21] Botong Wang. Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics.
arXiv e-prints arXiv:1509.03762, 2015.

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