The Chevalley–Warning Theorem is a result on the solution set of a system of polynomial equations $f_1,\dots ,f_r$ in $n$ variables over a finite field ${\mathbb{F}}_q$ in the low degree case $d≔{\sum }_{j=1}^{r}deg\left(f_j\right)<n$. In this note we reformulate that result in terms of fibers of the associated polynomial map and, following Heath-Brown, show that something weaker continues to hold when $d=n$. This result invites a search for homogeneous degree $n$ polynomials in $n$ variables over ${\mathbb{F}}_q$ for which the associated polynomial function ${\mathbb{F}}_q^n\to {\mathbb{F}}_q$ is not surjective, and we exhibit several families of such polynomials.

中文翻译：

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Chevalley-边界警告

Chevalley-Warning 定理是多项式方程组的解集的结果 $F_1,\dots ,F_r$ 在 $n$ 有限域上的变量 ${\mathbb{F}}_q$ 在低度的情况下 $d≔{\sum }_{j=1}^r度数\left(F_j\right)<n$. 在本笔记中，我们根据相关多项式映射的纤维重新表述该结果，并遵循 Heath-Brown，表明当较弱的东西继续成立时$d=n$. 此结果邀请搜索同质度$n$ 多项式 $n$ 变数结束 ${\mathbb{F}}_q$ 其中相关的多项式函数 ${\mathbb{F}}_q^n\to {\mathbb{F}}_q$ 不是满射的，我们展示了几个这样的多项式族。