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Let $C\subset{\mathbb P}_K^2$ be an algebraic curve over a number field $K$,and denote by $d_K$ the degree of $K$ over ${\mathbb Q}$. We prove that thenumber of $K$-rational points of height at most $H$ in $C$ is bounded by $cd^{2}H^{2d_K/d}(\log H)^\kappa$ where $c,\kappa$ are absolute constants. Wealso prove analogous results for global fields in positive characteristic, and,for higher dimensional varieties.  The quadratic dependence on $d$ in the bound as well as the exponent of $H$are optimal; the novel aspect is the quadratic dependence on $d$ which answersa question raised by Salberger. We derive new results on Heath-Brown's andSerre's dimension growth conjecture for global fields, which generalize inparticular the results by the first two authors and Novikov from the case$K={\mathbb Q}$. The proofs however are of a completely different nature,replacing the real analytic approach previously used by the $p$-adicdeterminant method. The optimal dependence on $d$ is achieved using a technicalimprovement in the treatment of high multiplicity points on mod $p$ reductionsof algebraic curves.

Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields