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- W2471925304 abstract "We show that for any sufficiently large integer $Q$ and a real $0leqlambdaleqfrac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,lambda)={(x,y):|y-f(x)|<Q^{-lambda}, xin J=[a,b]}$ 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 $°Pleq n, nge 2$, 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 $°Pleq n$ and height $H(P)leq Q$. This theorem is a generalization of a result obtained by V. Bernik, F. Gotze and O. Kukso for $s_1=s_2=frac12$ and $lambda = frac12$." @default.
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- W2471925304 date "2016-02-04" @default.
- W2471925304 modified "2023-09-27" @default.
- W2471925304 title "On points with algebraically conjugate coordinates close to smooth curves" @default.
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