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

Given $A\subseteq \Z$, 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.

For a given form, we investigate whether this happens for all but finitely many $p$.

We consider the more general question when forms have coefficients in Dedekind domains, and, under certain conditions, we prove that the analogous statement holds.

However, we also give examples of integral forms for which the answer is negative, and it is crucial that the degree of such forms is composite.

We conjecture that we cannot find such examples with forms of prime degree having sufficiently many variables, which is indeed the case when the degree is 2, 3, or 5.

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.

We construct integral forms such that the ratio set of its values is dense in at least one $\Q_p$ but only in finitely many of them.