Rational points on complete symmetric hypersurfaces over finite fields
Rational points on complete symmetric hypersurfaces over finite fields
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DOI：
10.1016/j.disc.2020.112072
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发表时间：
2020-07
期刊：
影响因子：
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通讯作者：
Jun Zhang;D. Wan
中科院分区：
文献类型：
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作者：
Jun Zhang;D. Wan
对于q元有限域Fq上m次m次k≥3元的完全对称多项式所定义的仿射超曲面，如果1−m≤q≤3且q为奇数，则该超曲面在Fq上至少有6 q k−3个有理点.我们证明的一个关键部分是Segre关于有限射影平面上的椭圆的经典定理。
For any affine hypersurface defined by a complete symmetric polynomial in k≥ 3 variables of degree m over the finite field F q of q elements, a special case of our theorem says that this hypersurface has at least 6 q k− 3 rational points over F q if 1≤ m≤ q− 3 and q is odd. A key ingredient in our proof is Segre’s classical theorem on ovals in finite projective planes.