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September 20, 2025Forum of Mathematics SigmaOpen Access

Improvements on dimension growth results and effective Hilbert’s irreducibility theorem

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Research demonstrates refined dimension growth bounds for algebraic varieties, suggesting new insights on Hilbert’s irreducibility theorem.

Dimension growth bounds are sharpened for irreducible algebraic varieties, allowing better estimation of points with bounded height.
The study focuses on affine hypersurfaces, generalizing previous results and refining the dependence on polynomial degree.
A conjecture is introduced regarding plane curves, aiming to resolve the uniform degree 3 case that remains open.
An effective higher-dimensional version of Hilbert’s irreducibility theorem is developed, highlighting its independent significance.

Cluckers et al. (2025) studied this question.

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Research demonstrates refined dimension growth bounds for algebraic varieties, suggesting new insights on Hilbert’s irreducibility theorem.