  We present sharp estimates for the degree and the height of the polynomials
in the Nullstellensatz over $\Z$. The result improves previous work of
Philippon, Berenstein-Yger and Krick-Pardo.
  We also present degree and height estimates of intrinsic type, which depend
mainly on the degree and the height of the input polynomial system. As an
application, we derive an effective arithmetic Nullstellensatz for sparse
polynomial systems.
  The proof of these results relies heavily on the notion of local height of an
affine variety defined over a number field. We introduce this notion and study
its basic properties.
