Improvements on dimension growth results and effective Hilbert's irreducibility theorem
arxiv(2023)
摘要
We sharpen and generalize the dimension growth bounds for the number of
points of bounded height lying on an irreducible algebraic variety of degree
$d$, over any global field. In particular, we focus on the the affine
hypersurface situation by relaxing the condition on the top degree homogeneous
part of the polynomial describing the affine hypersurface. Our work sharpens
the dependence on the degree in the bounds, compared to~\cite{CCDN-dgc}. We
also formulate a conjecture about plane curves which gives a conjectural
approach to the uniform degree $3$ case (the only case which remains open). For
induction on dimension, we develop a higher dimensional effective version of
Hilbert's irreducibility theorem.
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