有限域F_q上一类超曲面上zeta函数的计算

Computing zeta functions of a hypersurface over finite field F_q

设F=Fq是一个q元有限域,其中q=pf,f≥1,p是一个奇素数.利用有限域F=Fq上一类方程:a1xd111...xd1,m+m1+1+a2xd121...xd2,m+m1+1xd2,m+2m+2+...+akxd1k1...xdk,m+m1+1...xdkm,+km+k=0,其中m≥0,k≥1,dij≥0,ai∈F*,b∈F当指数满足一定条件时,在(F*)m+k上解数的直接公式结果,给出相应射影簇的zeta函数的可计算公式.最后,应用这些公式计算了一具体方程的zeta函数.

Let F be a finite field with q = p^f elements,where p is a prime number. The authors applied the explicit formula for the number of solutions of the following equation to obtain several formulas of it＇ s zeta function： a1x1^d11…xm＋1^d1.m＋1＋a2x1^d21…xm＋1^d2.m＋1xm＋2^d2.m＋2＋…akx1^dk1…xm＋k^dk.m＋k=0, in F^q where m≥0,k≥1,djj≥0,ai∈F^＊,b∈F,（d1,m＋1,d2,m＋2,…,dk,m＋k,q-1）=1. Finally, the authors applied thoseformulas to compute the zeta function of a given equation.