Improvements upon the Chevalley–Warning–Ax–Katz-type estimates☆
Improvements upon the Chevalley–Warning–Ax–Katz-type estimates☆
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DOI：
10.1016/j.jnt.2006.04.003
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发表时间：
2007
影响因子：
0.7
通讯作者：
W. Cao;Qi Sun
中科院分区：
文献类型：
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作者：
W. Cao;Qi Sun
设f是有限域Fq上的q元多项式，N（f=0）表示f在Fq中的零点个数. N（f=0）的q-整除性已被许多学者研究过，如Chavelley，Warning，Ax，Katz等.本文通过降低多项式的次数而不改变其零点个数，在许多情况下改进了Chevalley-Warning-Ax-Katz型估计.此外，我们的结果改进了Cao-Sun最近关于有限域上一般对角方程零点个数的约化。
Let f be a polynomial over finite field Fqwith q elements and let N(f=0) denote the number of zeros of f in Fq. The q-divisibility properties of N(f=0) have been studied by many authors, such as Chavelley, Warning, Ax, Katz, etc. In this paper, by reducing the degree of a given polynomial meanwhile remaining the number of its zeros unchanged, we present an improvement upon the Chevalley–Warning–Ax–Katz-type estimates in many cases. Furthermore, our result can improve Cao–Sun's reduction recently obtained on counting the number of zeros of general diagonal equations over finite fields.