After a short review of the history of this problem, I will sketch its relationship with some aspects of diophantine approximation and complexity theory.
Our treatement of the arithmetic Nullstellensatz is based on duality theory for Gorenstein algebras (Tate trace formula). The trace formula allows one to perform division modulo complete intersection ideals, with good control of the degree and height of the involved polynomials.
The key ingredient is the notion of local height of a variety defined over a number field. The local arithmetic intersection theory plays --- with respect to the height estimates --- the role of the classical intersection theory with respect to the degree bounds.
(Joint work with T. Krick (Univ. Buenos Aires, Argentina) and L. M. Pardo (Univ. Cantabria, Spain).)