论文标题
在有限场上完全对称超出曲面上的理性点
Rational points on complete symmetric hypersurfaces over finite fields
论文作者
论文摘要
对于由$ k \ geq 3 $ $ m $变量$ \ mathbb {f} _ {q} $ of $ q $元素的完全对称的多项式定义的任何仿射高度表面，我们的特殊案例说，我们的特殊案例说，这种高度的hypersurface表示至少有6q^{k-3} $ 6Q^{k-3} $ rations $ \ mathbb {f} _ {q} $如果$ 1 \ leq m \ leq q-3 $和$ q $是奇数。我们证明的关键要素是Segre在有限的投影平面中的椭圆形经典定理。
For any affine hypersurface defined by a complete symmetric polynomial in $k\geq 3$ variables of degree $m$ over the finite field $\mathbb{F}_{q}$ of $q$ elements, a special case of our theorem says that this hypersurface has at least $6q^{k-3}$ rational points over $\mathbb{F}_{q}$ if $1\leq m \leq q-3$ and $q$ is odd. A key ingredient in our proof is Segre's classical theorem on ovals in finite projective planes.
