November  2014, 8(4): 497-509. doi: 10.3934/amc.2014.8.497

On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$

1. 

Équipe GAATI, Université de la Polynésie Française, BP 6570, 98702 FAA'A, Tahiti, Polynésie Française, France

Received  January 2014 Revised  September 2014 Published  November 2014

Let $t$ be an integer $\ge 5$. The absolute irreducibility of the polynomial $\phi_t(x, y) = \frac{x^t + y^t + 1 + (x + y + 1)^t}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb{F}_2$) plays an important role in the study of APN functions. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.
Citation: Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
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show all references

References:
[1]

in Finite Fields: Theory and Applications, Amer. Math. Soc., Providence, 2010, 23-31. doi: 10.1090/conm/518/10193.  Google Scholar

[2]

IEEE Trans. Inf. Theory, 52 (2006), 4160-4170. doi: 10.1109/TIT.2006.880036.  Google Scholar

[3]

Finite Fields Appl., 24 (2013), 118-123. doi: 10.1016/j.ffa.2013.06.003.  Google Scholar

[4]

Finite Fields Appl., 14 (2008), 703-714. doi: 10.1016/j.ffa.2007.11.002.  Google Scholar

[5]

Finite Fields Appl., 25 (2014), 26-36. doi: 10.1016/j.ffa.2013.08.006.  Google Scholar

[6]

in Proc. IEEE Int. Symp. Inf. Theory, 2006, 2637-2641. doi: 10.1109/ISIT.2006.262131.  Google Scholar

[7]

in Proc. Workshop Coding Crypt., 2005, 316-324. Google Scholar

[8]

Des. Codes Crypt., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.  Google Scholar

[9]

Des. Codes Crypt., 73 (2014), 601-614. doi: 10.1007/s10623-014-9956-2.  Google Scholar

[10]

M. Delgado and H. Janwa, On the conjecture on APN functions,, preprint, ().   Google Scholar

[11]

invited talk at 9th Int. Conf. Finite Fields Appl., 2009. Google Scholar

[12]

IEEE Trans. Inf. Theory, 52 (2006), 744-747. doi: 10.1109/TIT.2005.862128.  Google Scholar

[13]

in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2012, 27-36. doi: 10.1090/conm/574/11423.  Google Scholar

[14]

in Algebraic Geometry and its Applications, World Sci. Publ., Hackensack, 2008, 388-409. doi: 10.1142/9789812793430_0021.  Google Scholar

[15]

in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2010, 41-53. doi: 10.1090/conm/521/10272.  Google Scholar

[16]

Benjamin, New York, 1969.  Google Scholar

[17]

Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511755224.  Google Scholar

[18]

J. Algebra, 343 (2011), 78-92. doi: 10.1016/j.jalgebra.2011.06.019.  Google Scholar

[19]

J. Algebra, 178 (1995), 665-676. doi: 10.1006/jabr.1995.1372.  Google Scholar

[20]

in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (eds. G. Cohen, T. Mora and O. Moreno), Springer-Verlag, NewYork, 1993, 180-194. doi: 10.1007/3-540-56686-4_43.  Google Scholar

[21]

Finite Fields Appl., 13 (2007), 1006-1028. doi: 10.1016/j.ffa.2007.04.004.  Google Scholar

[22]

Ph.D thesis, Université Paris 7, 2011; available online at http://www.math.u-psud.fr/ leducq/these.pdf Google Scholar

[23]

Amer. J. Math., 1 (1878), 197-240 and 289-321. doi: 10.2307/2369373.  Google Scholar

[24]

in Adv. Crypt.- Eurocrypt '93, Springer, Berlin, 1994, 55-64. doi: 10.1007/3-540-48285-7_6.  Google Scholar

[25]

in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 169-181. doi: 10.1090/conm/487/09531.  Google Scholar

[26]

Crypt. Commun., 3 (2011), 227-240. doi: 10.1007/s12095-011-0050-6.  Google Scholar

[27]

The Sage Development Team, Sage Mathematics Software (Version 4.8),, , ().   Google Scholar

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