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

V. Bernik ; F. Götze ; A. Gusakova

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 14-47

Résumé

Let $\varphi\colon\mathbb R\to\mathbb R$ be a continuously differentiable function on a finite interval $J\subset\mathbb R$, and let $\boldsymbol\alpha=(\alpha_1,\alpha_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $\alpha_1,\alpha_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_\varphi(Q,\gamma,J)$ the set of points $\boldsymbol\alpha$ such that $|\varphi(\alpha_1)-\alpha_2|\leq c_1Q^{-\gamma}$. We show that for $0\gamma1$ and any sufficiently large $Q$ there exist positive values $c_2$, where $c_i=c_i(n)$, $i=1,2$, that are independent of $Q$ and such that $c_2\cdot Q^{n+1-\gamma}\# M^n_\varphi(Q,\gamma,J)$.