On points with algebraically conjugate coordinates close to smooth curves

Abstract

We show that for any sufficiently large integer $Q$ and a real $0\le \lambda \le \frac{3}{4}$ 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 $°P\le n,n\ge 2$, and height $H(P)\le Q$. 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 $|P^{\prime }(x)|,|P^{\prime }(y)|$ bounded from below, where $P(x)$ is a polynomial of degree $°P\le n$ and height $H(P)\le Q$. This theorem is a generalization of a result obtained by V. Bernik, F. Götze and O. Kukso for $s_1=s_2=\frac{1}{2}$ and $\lambda =\frac{1}{2}$.

arXiv admin note: text overlap with arXiv:1608.00873

Cited by in corpus (2)

• On distribution of points with algebraically conjugate coordinates in neighborhood of smooth curves
• On algebraic integers in short intervals and near smooth curves