有限域上由两个广义对角多项式所确定的簇中的有理点

Rational points on the variety defined by two generalized diagonal polynomials over finite field

设F_q为有限域,f_l=a_l1…+a_l1…+…+a_ln…+c_l(l=1,2)为F_q上的一组广义对角多项式,用N_q(V)表示由f_l(l=1,2)确定的族中的F_q有理点的个数.作者利用Adolphson和Sperber的牛顿多面体理论与指数和工具,证明了ord_qN_q(V)≥max{∑_i=1~n 1/d_i—2,0},其中d_i=max{d_ij^(1),d_ij^(2)|1≤j≤k_i},1≤i≤n.

Let F_q be the finite field and N_q(V) denote the number of F_q rational points on the variety determined by f_l=a_(l1)x_(11)^(d_(11)^(l))…x_(1k_1)^(d_(1k_1)^(l))+a_(l2)x_(21)^(d_(21)^(l))…x_(2k_2)^(d_(2k_2)^(l))+…+a_(ln)x_(n1)^(d_(n1)^(l))…x_(nk_n)^(d_(nk_n)^(l))+c_l(l=1,2).By using the Newton polyhedra technique introduced by Adolphson and Sperber,the authors prove that or d_qN_q(V)≥ max{‘Σ_(i=1)~n 1/(d_i)’-2,0},where d_i=max{d_(ij)^((1)),d_(ij)^((2))|1≤j≤k_i},1≤i≤n.