Number of $\mathbb{F}_q$-points on Diagonal hypersurfaces and hypergeometric function Article Swipe

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YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2210.11732

Let $D_λ^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$, $h_i\geq1$, $\sum_{i=1}^n h_i=d$, and $\gcd(d,h_1,h_2,\ldots, h_n)=1$. The Dwork hypersurface is the case when $d=n$, that is, $h_1=h_2=\cdots =h_n=1$. Formulas for the number of $\mathbb{F}_q$-points on the Dwork hypersurfaces in terms of McCarthy's $p$-adic hypergeometric functions are known. In this article we provide a formula for the number of $\mathbb{F}_q$-points on $D_λ^d$ in terms of McCarthy's $p$-adic hypergeometric function which holds for $d\geq n$.

Related Topics

Combinatorics

Hypergeometric Distribution

Mathematics

Gravitational Singularity

Physics

Mathematical Analysis

Geometry

Evolutionary Biology

Biology

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Hypersurface Monomial Diagonal Hypergeometric function Combinatorics Hypergeometric distribution Mathematics Finite field Field (mathematics) Gravitational singularity Function (biology) Physics Pure mathematics Mathematical analysis Geometry Evolutionary biology Biology

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2210.11732
PDF: https://arxiv.org/pdf/2210.11732
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Cited By: 1
Related Works: 10
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DOI: https://doi.org/10.48550/arxiv.2210.11732

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Title: Number of $\mathbb{F}_q$-points on Diagonal hypersurfaces and hypergeometric function

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Type: preprint

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Language: en

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Publication year: 2022

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Publication date: 2022-10-21

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Authors: Sulakashna, Rupam Barman

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Concepts: Hypersurface, Monomial, Diagonal, Hypergeometric function, Combinatorics, Hypergeometric distribution, Mathematics, Finite field, Field (mathematics), Gravitational singularity, Function (biology), Physics, Pure mathematics, Mathematical analysis, Geometry, Evolutionary biology, Biology

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