Hello all, let $n$ be an integer $\geq 2$ and let $\alpha$ be an algebraic number of degree $n$. Let $R$ be the ring of algebraic integers in ${\mathbb Q}(\alpha)$, and let $B$ be the subset of $R$ containing the elements whose degree is exactly $n$. Any $\beta \in B$ has a minimal polynomial $X^n+b_{n-1}X^{n-1}+ \ldots + b_1X+b_0$. Identifying this latter polynomial with the uple $(b_0,b_1, \ldots ,b_{n-1})$ allows us to view $B$ as a subset of ${\mathbb Z}^n$. I define a combinatorial subvariety $V$ of dimension at most $r$ of ${\mathbb Z}^n$ to be a subset of $Z^n$ such that there is a set of indices $I \subseteq \lbrace 1,2, \ldots , n \rbrace$ with $|I|=n-r$ and the projection $p:V \to {\mathbb Z}^{n-r}, (v_1,v_2, \ldots ,v_n) \mapsto (v_i)_{i\in I}$ is constant.

My question is : what is the smallest $r$ such that there is an infinite subset $B' \subset B$ corresponding to a subvariety of dimension at most $r$ ?

In other words, we are asking for infinitely many elements in $B$, whose minimal polynomials are ``as similar as possible".

An easy case is when $\alpha=a^{\frac{1}{n}}$ for some $a \in {\mathbb Q}$, because the rational multiples of $\mathbb \alpha$ correspond to a subvariety of dimension 1, so that $r=1$ in this case.