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

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  • 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.
    Mathematics Subject Classification: 11T06, 12E05, 14Q10, 11T71.


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    The Sage Development Team, Sage Mathematics Software (Version 4.8), http://www.sagemath.org

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