Sylvester-范德蒙德行列式最小幂非剩余的平方根界
Square root Bound on the Least Power Non-residue using a Sylvester-Vandermonde Determinant
摘要：
给出了素数域$\F_p$上等差数列$\{bn+c\}_{n=0,1...}$的最小$k^{th}$幂非剩余的值有界于$7/\sqrt{5} \cdot b \cdot \sqrt{p/k} + 4b + c$的一个新的初等证明.我们的证明受到所谓的\emph{Stepanov method}的启发，它涉及通过构造一个非零的低次辅助多项式来限制一个方程组的解集的大小，该辅助多项式在解集上具有高的多重性。证明使用了基本的代数和数论，以及一个推广了西尔维斯特和范德蒙德行列式的行列式恒等式。
英文摘要：
We give a new elementary proof of the fact that the value of the least $k^{th}$ power non-residue in an arithmetic progression $\{bn+c\}_{n=0,1...}$, over a prime field $\F_p$, is bounded by $7/\sqrt{5} \cdot b \cdot \sqrt{p/k} + 4b + c$. Our proof is inspired by the so called \emph{Stepanov method}, which involves bounding the size of the solution set of a system of equations by constructing a non-zero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number theory along with a determinant identity that generalizes both the Sylvester and the Vandermonde determinant.
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