Counting polynomials with distinct zeros in finite fields

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• 作者：Haiyan Zhou^a ; ^{haiyanxiaodong@gmail.com} ; Li-Ping Wang^b ; ^{wangliping@iie.ac.cn} ; Weiqiong Wang^c ; ^d ; ^{wqwang@chd.edu.cn}
• 关键词：Polynomials ; Inclusion&ndash ; exclusion principle ; Moments subset-sum ; Distinct coordinate sieve ; Spectrum of graphs
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：118-135
• 全文大小：370 K
• 卷排序：174

文摘

Let FqFq be a finite field with q=peq=pe elements, where p   is a prime and e≥1e≥1 is an integer. Let ℓ,nℓ,n be two positive integers such that ℓ<nℓ<n. Fix a monic polynomial u(x)=xn+un−1xn−1+⋯+uℓ+1xℓ+1∈Fq[x]u(x)=xn+un−1xn−1+⋯+uℓ+1xℓ+1∈Fq[x] of degree n and consider all degree n monic polynomials of the formf(x)=u(x)+vℓ(x),vℓ(x)=aℓxℓ+aℓ−1xℓ−1+⋯+a1x+a0∈Fq[x]. For any non-negative integer k≤min{n,q}k≤min{n,q}, let Nk(u(x),ℓ)Nk(u(x),ℓ) denote the total number of vℓ(x)vℓ(x) such that u(x)+vℓ(x)u(x)+vℓ(x) has exactly k   distinct roots in FqFq, i.e.Nk(u(x),ℓ)=|{f(x)=u(x)+vl(x)|f(x)has exactlykdistinct zeros inFq}|. In this paper, we obtain explicit combinatorial formulae for Nk(u(x),ℓ)Nk(u(x),ℓ) when n−ℓn−ℓ is small, namely when n−ℓ=1,2,3n−ℓ=1,2,3. As an application, we define two kinds of Wenger graphs called jumped Wenger graphs and obtain their explicit spectrum.