Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Complexity of the normalization of algebras by Wolmer VASCONCELOS, Rutgers University Abstract: Let R be a normal unmixed integral domain and let A be a semistandard graded R-algebra of integral closure \overline{A}. Estimating the number of steps that general algorithms must take to build \overline{A} can be viewed as an invariant of A. We show how the degree function jdeg(\cdot) can be used to provide bounds that depend on \overline{A}. In two major cases, algebras that allow Noether normalizations (affine algebras over fields or Z) and Rees algebras of ideals or modules, such estimates are related to invariants of A, in particular they can be said to be known ab initio.