arXiv 1809.062712018-09-17math.COmath.AG

特征为2的有限域上的低次平面多项式

Low-degree planar polynomials over finite fields of characteristic two

Daniele Bartoli, Kai-Uwe Schmidt

摘要：

平面函数是从有限域$\mathbb{F}_q$到其自身的具有极值微分性质的映射。这样的函数产生有限的射影平面和其他组合对象。有一个微妙的区别，这些功能的定义取决于奇偶性的$q$，我们认为的情况下，$q$甚至。我们分类多项式的程度最$q^{1/4}$诱导平面函数$\mathbb{F}_q$，通过显示，这样的多项式正是那些在其中的每一个单项式的程度是一个权力的2。作为推论，我们得到了例外平面多项式的一个完整分类，即在$\mathbb{F}_q$上的多项式，它在~ $\mathbb{F}_q$的无穷多个扩张上导出平面函数。证明策略是研究一条代数曲线与一个假定的平面函数相连的$\mathbb{F}_q$ -有理点的个数。我们的方法也给出了几乎完全非线性函数分类的一个新的部分结果的简单证明。

英文摘要：

Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of $q$ and we consider the case that $q$ is even. We classify polynomials of degree at most $q^{1/4}$ that induce planar functions on $\mathbb{F}_q$, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over $\mathbb{F}_q$ that induce planar functions on infinitely many extensions of~$\mathbb{F}_q$. The proof strategy is to study the number of $\mathbb{F}_q$-rational points of an algebraic curve attached to a putative planar function.~Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear~functions.

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摘要：

英文摘要：

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