有限域上完全对称多项式零点的新下界  

New Bounds for Zeros of Complete Symmetric Polynomials over Finite Fields

作　　者：万大庆 张俊 Da Qing WAN;Jun ZHANG(Department of Mathematics,University of California,Irvine,CA 92697,USA;School of Mathematical Sciences,Capital Normal University,Beijing 100048,P.R.China)

机构地区：[1]美国加州大学尔湾分校数学系,尔湾CA 92697 [2]首都师范大学数学科学学院,北京100048

出　　处：《数学学报（中文版）》2024年第2期211-219,共9页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11971321,12222113);科技部重点研发计划(2018YFA0704703)。

摘　　要：有限域上多项式的零点计数问题是算术代数几何的核心问题之一,本文考虑有限域Fq上完全对称多项式的零点问题.主要结果如下:设h(x1,…,xk)是有限域Fq上一个m次完全对称多项式(k≥3,1≤m≤q-2):(1)若q为奇数,则h(x1,…,xk)在Fqk中至少有[(q-1)/(m+1)]/(q-[(q-1)/(m+1)])(q-m-1)q^(k-2)个零点;(2)若q为偶数,且k≥4,则h(x1,…,xk)在Fqk中至少有[(q-1)/(m+1)]/(q-[(q-1)/(m+1)])(q-(m+1)/2)(q-1)q^(k-3)个零点.注意到,当m比较小的时候,上述新的下界改进了已有下界[4,定理1.4]和[3,定理1.2](见本文结论1.1和1.2)大约q2/6m倍.Counting zeros of polynomials over finite fields is one of the most important topics in arithmetic algebraic geometry.In this paper,we consider the problem for complete symmetric polynomials.The homogeneous complete symmetric polynomial of degree m in the k-variables{x_1,x_2,...,x_k}is defined to be h_m(x_1,x_2,…,x_k):=∑_(1≤i_1≤i_2≤…≤i_m≤k)x_(i_1)x_(i_2)…x_(i_m).A complete symmetric polynomial of degree m over F_q in the k-variables{x_1,x_2,…,x_k}is defined to be h(x_1,…,x_k):=∑_(e=0)~m a_eh_e(x_1,x_2,…,x_k),where a_e∈F_q and a_m≠0.Let N_q(h):=#{(x_1,...,x_k)∈F_q~k|h(x_1,…,x_k)=0}denote the number of F_q-rational points on the affine hypersurface defined by h(x_1,…,x_k)=0.In this paper,we improve the bounds given in[J.Zhang and D.Wan,"Rational points on complete symmetric hypersurfaces over finite fields",Discrete Mathematics,343(11):112072,2020]and[D.Wan and J.Zhang,"Complete symmetric polynomials over finite fields have many rational zeros"Scientia Sinica Mathematica,51(10):1677-1684,2021].Explicitly,we obtain the following new bounds:(1)Let h(x_1,…,x_k)be a complete symmetric polynomial in k≥3 variables over F_q of degree m with 1≤m≤q-2.If q is odd,then N_q(h)≥[(q-1)/(m+1)/(q-[(q-1)/m+1)])(q-m-1)q~(k-2).(2)Let h(x_1,…,x_k)be a complete symmetric polynomial in k>4 variables over F_q of degree m with 1≤m≤q-2.If q is even,then N_q(h)≥[(q-1)/(m+1)]/(q-[(q-1)/m+1)])(q-(m+1)/2)(q-1)q~(k-3).Note that our new bounds roughly improve the bounds mentioned in the above two papers by the factor q~2/6m for small degree m.

关 键 词：完全对称多项式 零点 有限域 

分 类 号：O156[理学—数学]