On distribution of points with algebraically conjugate coordinates in neighborhood of smooth curves

Abstract

Let $\phi :\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function on an interval $J\subset \mathbb{R}$ and let $\mathbf{\alpha }=({\alpha }_1,{\alpha }_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of ${\alpha }_1,{\alpha }_2$ is of degree $\le n$ and height $\le Q$. Denote by $M_{\phi }^n(Q,\gamma ,J)$ the set of such points $\mathbf{\alpha }$ such that $|\phi ({\alpha }_1)-{\alpha }_2|\le c_1{Q}^{-\gamma }$. We show that for a real $0<\gamma <1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, where $c_i=c_i(n)$, $i=1,2$, which are independent of $Q$, such that $c_2{Q}^{n+1-\gamma }<\#M_{\phi }^n(Q,\gamma ,J)<c_3{Q}^{n+1-\gamma }$.

References in corpus (1)

• On points with algebraically conjugate coordinates close to smooth curves