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Fkq中至少有「q-1/m+1(」)/q-「q-1/m+1(「)(q-m-1)qk-2个零点;(2)若q为偶数,且k≥4,则h(x1,…,xk)在Fkq中至少有「q-1/m+1(「)/q-「q-1/m+1(」)(q-m+1/2)(q-1)qk-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.
New Bounds for Zeros of Complete Symmetric Polynomials over Finite Fields
Counting zeros of polynomials over finite fields is one of the most impor-tant 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 {x1,x2,…,xk} is defined to be hm(x1,x2,…,xk):=∑1≤i1≤i2≤…≤im≤kxi1xi2…xim.A complete symmetric polynomial of degree m over Fq in the k-variables {x1,x2,…,xk} is defined to be h(x1,…,xk):=∑me=0aehe(x1,x2…,xk),where ae ∈ Fq and am≠0.Let Nq(h):=#{(x1,…,xk)∈Fkq|h(x1,…,xk)=0} denote the number of Fq-rational points on the affine hypersurface defined by h(x1,…,xk)=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(x1,…,xk)be a complete symmetric polynomial in k ≥ 3 variables over Fq of degree m with 1 ≤ m ≤ q-2.If q is odd,then Nq(h)≥「q-1/m+1(」)/q-「q-1/m+1(」)(q-m-1)qk-2.(2)Let h(x1,…,xk)be a complete symmetric polynomial in k ≥ 4 variables over Fq of degree m with 1 ≤ m ≤ q-2.If q is even,then Nq(h)≥「q-1/m+1(」)/q-「q-1/m+1(」)(q-m+1/2)(q-1)qk-3.Note that our new bounds roughly improve the bounds mentioned in the above two papers by the factor q2/6m small degree m.
complete symmetric polynomialrational zerofinite field
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