p-例外单项式GAPN函数的几何刻画
Geometric characterization of $p$-exceptional monomial GAPN functions
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中文总结 AI 辅助
本文针对p-例外单项式GAPN函数，通过建立有限域上二元多项式的几何刻画，解决了该类函数分类的公开问题。
中文摘要 AI 辅助
在特征为p的有限域上，奇素数p对应的PN（完全非线性）函数、偶素数p对应的APN（几乎完全非线性）函数是两类著名的高度非线性函数。GAPN（广义几乎完全非线性）函数是将偶素数p的APN函数推广至所有素数p的函数类。这类高度非线性函数研究的核心目标之一是其分类：p-例外单项式PN函数与2-例外单项式APN函数已完成分类，但GAPN函数的对应问题仍未解决。本文定义：若F_p上的多项式对无穷多个正整数n，均为F_{p^n}上的PN（或APN、或GAPN）函数，则称其为p-例外PN（或APN、或GAPN）函数。为此，本文对素数幂q，建立了F_q[x,y]中无F_q上定义的绝对不可约因子的非零多项式的几何刻画，进而给出p-例外单项式GAPN函数的几何刻画。
英文摘要
On finite fields of characteristic $p$, PN (perfect nonlinear) functions for odd $p$ and APN (almost perfect nonlinear) functions for even $p$ are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even $p$ to all $p$. One of the main targets of studies on such highly nonlinear functions is their classification. While $p$-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over $\mathbb{F}_p$ is called a $p$-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on $\mathbb{F}_{p^n}$ for infinitely many positive integers $n$. In this paper, we give a geometric characterization of $p$-exceptional monomial GAPN functions. To this end, for a prime power $q$, we establish a geometric characterization of non-zero polynomials in $\mathbb{F}_{q}[x,y]$ having no absolutely irreducible factor defined over $\mathbb{F}_{q}$.
