Defect of an extension, key polynomials and local uniformization
  For all simple and finite extension of a valued field, we prove that its defect is the product of the effective degrees of the complete set of key polynomials associated. As a consequence, we obtain a local uniformization theorem for valuations of rank 1 centered on an equicharacteristic quasi-excellent local domain satisfying some inductive assumptions of lack of defect. 
Jean-Christophe San Saturnino
math.AG