On points with algebraically conjugate coordinates close to smooth curves

Abstract

We show that for any sufficiently large integer $Q$ and a real $0\leq\lambda\leq\frac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,\lambda)=\{(x,y):|y-f(x)|<Q^{-\lambda}, x\in J=[a,b]\}$ contain at least $c(n, f, J)Q^{n+1-\lambda}$ points $\bar{\gamma}=(\alpha,\beta)$ with algebraically conjugate coordinates. We consider points $\bar{\gamma}$ such that the minimal polynomial $P(x)$ of $\alpha,\beta$ is of degree $\deg P\leq n,\ n\ge 2$, and height $H(P)\leq Q$. The proof is based on a metric theorem on the measure of the set of vectors $(x,y)$ lying in a rectangle $\Pi$ of dimensions $Q^{-s_1}\times Q^{-s_2}$ with $|P(x)|, |P(y)|$ bounded from above and $|P'(x)|,|P'(y)|$ bounded from below, where $P(x)$ is a polynomial of degree $\deg P\leq n$ and height $H(P)\leq Q$. This theorem is a generalization of a result obtained by V. Bernik, F. G\"otze and O. Kukso for $s_1=s_2=\frac12$ and $\lambda = \frac12$.