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Uniform bounds for the number of rational points on varieties over global fields

### Marcelo Paredes and Román Sasyk

Vol. 16 (2022), No. 8, 1941–2000
##### Abstract

We extend the work of Salberger; Walsh; Castryck, Cluckers, Dittmann and Nguyen; and Vermeulen to prove the uniform dimension growth conjecture of Heath-Brown and Serre for varieties of degree at least $4$ over global fields. As an intermediate step, we generalize the bounds of Bombieri and Pila to curves over global fields and in doing so we improve the ${B}^{𝜀}$ factor by a $\mathrm{log}\left(B\right)$ factor.

##### Keywords
varieties over global fields, heights in global fields, number of rational solutions of diophantine equations, determinant method
##### Mathematical Subject Classification
Primary: 11D45, 11G35, 11G50, 14G05