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
查看参考文献4篇
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文摘
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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(x_1,...,x_k)是有限域Fq上一个m次完全对称多项式(k≥3, 1 ≤m≤q-2): ⑴若q为奇数,则h(x_1,...,x_k)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(x_1,…,x_k)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. |
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其他语种文摘
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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,x2,...,x_k} is defined to be hm(x_1,x2,..., x_k):= ∑1≤i1≤i2≤…≤im≤k xi1xi2 … χim. A complete symmetric polynomial of degree m over Fq in the k-variables {x_1,x2,..., x_k}, is defined to be h(x_1,…,x_k):= ∑e=0m aehe(x_1,x2,…,Xk), where ae ∈ Fq and am ≠ 0. Let N_q(h):= #{(x_1,...,x_k) ∈ Fqk| h(x_1,...,x_k)= 0} denote the number of Fq-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 Mathematical 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 Fq 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 Fq 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 q2/6m for small degree m. |
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来源
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数学学报
,2024,67(2):211-219 【核心库】
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DOI
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10.12386/A20220143
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关键词
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完全对称多项式
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零点
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有限域
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地址
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1.
美国加州大学尔湾分校数学系, 尔湾, CA92697
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首都师范大学数学科学学院, 北京, 100048
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语种
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中文 |
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文献类型
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研究性论文 |
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ISSN
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0583-1431 |
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学科
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数学 |
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基金
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国家自然科学基金资助项目
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科技部重点研发计划
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文献收藏号
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CSCD:7673872
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参考文献 共
4
共1页
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CSCD被引
26
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2.
Wan D Q. A p-adic lifting lemma and its applications to permutation polynomials.
Finite Fields, Coding Theory, and Advances in Communications and Computing (LasVegas, NV, 1991), Lecture Notes in Pure and Applied Mathematics, Vol. 141,1993:209-216
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CSCD被引
1
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3.
Wan D Q. 有限域上完全对称多项式有很多零点.
中国科学. 数学,2021,51(10):1677-1684
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CSCD被引
3
次
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4.
Zhang J. Rational points on complete symmetric hypersurfaces over finite fields.
Discrete Mathematics,2020,343(11):112072
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CSCD被引
4
次
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