Uniform Bounds in Algebraic Geometry and Commutative Algebra

In this survey article, we will introduce various measures of complexity for algebraic constructions in polynomial rings over fields and show how they are often uniformly bounded by the complexity of the starting data. In problems which have a linear nature, the degree of the polynomials provide a sufficient notion of complexity. However, in the non-linear case, the more sophisticated measure of etale complexity is needed. These bounds lead often to the constructible nature of geometric problems, where in the non-linear case, one should work in the etale site rather than in the Zariski site. As another application of the existence of these bounds we mention the possibility of transferring results from one characteristic to another by means of the Lefschetz Principle. We will give some examples of new results as well as some new proofs to old results.