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This paper focuses on incidences over finite fields, extending to higherdegrees a result by Vinh \cite{VINH20111177} on the number of point-lineincidences 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 andpolynomials of bounded degree in $\mathbb{F}^2$. Our approach employs asingular value decomposition of the incidence matrix between points andpolynomials, coupled with an analysis of the related group algebras. This boundis then applied to coding theory, specifically to the problem of average-radiuslist decoding of Reed-Solomon (RS) codes. We demonstrate that RS codes ofcertain 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 waspreviously established, its existence for the stronger notion of average-radiuslist-decoding was not known to exist.

Points-Polynomials Incidence Theorem with Applications to Coding Theory