$p$-Adic quotient sets: diagonal forms

TL;DR

This paper investigates the denseness of quotient sets in the $p$-adic numbers generated by nonzero values of diagonal forms of degree $n$, extending previous results from cubic to higher degrees.

Contribution

It generalizes the study of $p$-adic denseness of quotient sets from quadratic and cubic forms to all diagonal forms of degree $n extgreater=3$, under certain conditions.

Findings

Extended results to all diagonal forms $ax^n+by^n$ for $n extgreater=3$.

Established conditions for $p$-adic denseness when $ ext{gcd}(n,p(p-1))=1$.

Connected the problem to the properties of integral forms and their value sets in $p$-adic analysis.

Abstract

For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic denseness of quotients of nonzero values attained by diagonal forms of degree $n\geq 3$, where $\gcd(n,p(p-1))=1$.