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
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
factor
by a
factor.
Keywords
varieties over global fields, heights in global fields,
number of rational solutions of diophantine equations,
determinant method
Department of Mathematics
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria – Pabellón I
1428
Ciudad Autónoma de Buenos Aires
Argentina
Department of Mathematics
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria – Pabellón I
1428
Ciudad Autónoma de Buenos Aires
Argentina
Instituto Argentino de Matemáticas
Alberto P. Calderón-CONICET
Saavedra 15, Piso 3
1083
Ciudad Autónoma de Buenos Aires
Argentina