论文标题
$ p $ - 亚种商集：对角线形式
$p$-Adic quotient sets: diagonal forms
论文作者
论文摘要
对于一组整数$ a $，我们考虑$ r（a）= \ {a/b：a，b \ in A，b \ neq 0 \} $。研究$ r（a）$在$ p $ - adic数字中的密度（$ a $ a $是通过积分表格实现的非零值集时）是一个开放的问题。二次形式已经回答了此问题。最近，Antony和Barman研究了对角二进制立方体$ ax^3+by^3 $的问题，其中$ a $和$ b $是整数。在本文中，我们研究了对角线形式的这个问题。我们将Antony和Barman的结果扩展到所有$ n \ geq 3 $的对角线二进制形式$ ax^n+。我们还研究了$ p $ - ad的非零值的商的密度，由$ n \ geq 3 $的对角线形式获得，其中$ \ gcd（n，p（p-1））= 1 $。
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 $\gcd(n,p(p-1))=1$.
