This paper focuses on incidences over finite fields, extending to higher degrees a result by Vinh \cite{VINH20111177} on the number of point-line incidences in the plane $\mathbb{F}^2$, where $\mathbb{F}$ is a finite field. Specifically, we present a bound on the number of incidences between points and polynomials of bounded degree in $\mathbb{F}^2$. Our approach employs a singular value decomposition of the incidence matrix between points and polynomials, coupled with an analysis of the related group algebras. This bound is then applied to coding theory, specifically to the problem of average-radius list decoding of Reed-Solomon (RS) codes. We demonstrate that RS codes of certain lengths are average-radius list-decodable with a constant list size, which is dependent on the code rate and the distance from the Johnson radius. While a constant list size for list-decoding of RS codes in this regime was previously established, its existence for the stronger notion of average-radius list-decoding was not known to exist.

翻译：本文研究有限域上的关联问题，将Vinh关于平面$\mathbb{F}^2$中点线关联数的结果推广至更高次数（其中$\mathbb{F}$为有限域）。具体而言，我们给出了$\mathbb{F}^2$中点与有界次数多项式之间关联数的上界。该方法利用点-多项式关联矩阵的奇异值分解，并结合对相关群代数的分析。该上界随后应用于编码理论，特别针对Reed-Solomon（RS）码的平均半径列表译码问题。我们证明：特定长度的RS码在常数列表大小下可实现平均半径列表译码，该常数依赖于码率及与Johnson半径的距离。尽管在此参数范围内RS码列表译码的常数列表大小此前已被证明，但更强的平均半径列表译码概念下其存在性尚未被证实。