On $p$-adic denseness of quotients of values of integral forms

TL;DR

This paper investigates when the ratios of values of integral forms are dense in the p-adic numbers, proving results for forms with many variables and conjecturing broader cases, using algebraic geometry tools.

Contribution

It introduces algebraic geometry methods to study p-adic denseness of quotient sets of integral form values, providing new results and counterexamples.

Findings

For non-singular forms with ≥3 variables, ratio sets are dense in  for large p.

Conjecture: non-degenerate forms of prime degree with many variables have dense ratio sets in .

Counterexamples show limitations when the number of variables equals the degree or for composite degrees.

Abstract

Given $A\subseteq \mathbb{Z}$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained by an integral form. In this paper, we consider the question of whether this happens for all but finitely many $p$. We prove that if a form is non-singular and has at least three variables, then the ratio set of its values is dense in $\mathbb{Q}_p$ for all sufficiently large $p$. We conjecture that the same statement is true for non-degenerate forms of prime degree having sufficiently many variables, which is indeed the case when the degree is 2, 3, or 5. Still, we give two counterexamples when the assumptions of the conjecture are not satisfied. The first one happens when the number of variables is equal to the degree. The other works for any number of variables but only for composite degrees. Our innovation is to consider the problem in terms of varieties defined by forms, thereby using the tools from algebraic geometry, making their first appearance in this setting. Finally, we construct integral forms such that the ratio set of its values is dense in at least one $\mathbb{Q}_p$ but only in finitely many of them.